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A Short Course in Differential Topology
A Short Course in Differential Topology
Bjørn Ian Dundas
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Manifolds are abound in mathematics and physics, and increasingly in cybernetics and visualization where they often reflect properties of complex systems and their configurations. Differential topology gives us the tools to study these spaces and extract information about the underlying systems. This book offers a concise and modern introduction to the core topics of differential topology for advanced undergraduates and beginning graduate students. It covers the basics on smooth manifolds and their tangent spaces before moving on to regular values and transversality, smooth flows and differential equations on manifolds, and the theory of vector bundles and locally trivial fibrations. The final chapter gives examples of localtoglobal properties, a short introduction to Morse theory and a proof of Ehresmann's fibration theorem. The treatment is handson, including many concrete examples and exercises woven into the text, with hints provided to guide the student.
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Год:
2018
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Cambridge University Press
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english
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266
ISBN 10:
1108349137
ISBN 13:
9781108349130
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Cambridge Mathematical Textbooks
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PDF, 19,11 MB
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Ключевые фразы
bundle^{463}
vector^{416}
manifold^{227}
chart^{210}
tangent^{175}
theorem^{172}
linear^{151}
bundles^{145}
manifolds^{140}
isomorphism^{129}
charts^{127}
atlas^{126}
matrix^{121}
compact^{121}
topology^{118}
inverse^{109}
lemma^{108}
dimensional^{108}
spaces^{105}
continuous^{105}
trivial^{104}
subset^{93}
differential^{91}
fiber^{91}
vector bundle^{91}
rpn^{90}
diffeomorphism^{89}
vector bundles^{88}
projection^{78}
vector space^{78}
hom^{78}
maps^{77}
submanifold^{77}
smooth manifold^{75}
tangent space^{71}
define^{70}
composite^{69}
cotangent^{67}
locally^{65}
germ^{65}
topological^{62}
sphere^{61}
metric^{55}
smooth manifolds^{55}
tangent bundle^{54}
global^{52}
imbedding^{52}
dimension^{50}
neighborhood^{49}
euclidean^{48}
fibration^{47}
equivalence^{47}
cos^{46}
torus^{45}
equation^{45}
subspace^{45}
maximal^{44}
homeomorphism^{44}
velocity^{44}
appendix^{44}
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A Short Course in Differential Topology Manifolds abound in mathematics and physics, and increasingly in cybernetics and visu alization, where they often reflect properties of complex systems and their configurations. Differential topology gives us tools to study these spaces and extract information about the underlying systems. This book offers a concise and modern introduction to the core topics of differential topology for advanced undergraduates and beginning graduate students. It covers the basics on smooth manifolds and their tangent spaces before moving on to regular values and transversality, smooth flows and differential equations on manifolds, and the theory of vector bundles and locally trivial fibrations. The final chapter gives examples of localto global properties, a short introduction to Morse theory and a proof of Ehresmann’s fibration theorem. The treatment is handson, including many concrete examples and exercises woven into the text, with hints provided to guide the student. Bjørn Ian Dundas is Professor in the Mathematics Department at the University of Bergen. Besides his research and teaching he is the author of three books. When not doing mathematics he enjoys hiking and fishing from his rowboat in northern Norway. CAMBRIDGE MATHEMATICAL TEXTBOOKS Cambridge Mathematical Textbooks is a program of undergraduate and beginning graduatelevel textbooks for core courses, new courses, and interdisciplinary courses in pure and applied mathematics. These texts provide motivation with plenty of exercises of varying difficulty, interesting examples, modern applications, and unique approaches to the material. Advisory Board John B. Conway, George Washington University Gregory F. Lawler, University of Chicago John M. Lee, University of Washington John Meier, Lafayette College Lawrence C. Washington, University of Maryland, College Park A complete list of books in the series can be found at www.cambridge.org/mathematics. Recent titles include the following: Chance, Strategy, an; d Choice: An Introduction to the Mathematics of Games and Elections, S. B. Smith Set Theory: A First Course, D. W. Cunningham Chaotic Dynamics: Fractals, Tilings, and Substitutions, G. R. Goodson A Second Course in Linear Algebra, S. R. Garcia & R. A. Horn Introduction to Experimental Mathematics, S. Eilers & R. Johansen Exploring Mathematics: An Engaging Introduction to Proof, J. Meier & D. Smith A First Course in Analysis, J. B. Conway Introduction to Probability, D. F. Anderson, T. Seppäläinen & B. Valkó Linear Algebra, E. S. Meckes & M. W. Meckes A Short Course in Differential Topology Bjørn Ian Dundas Universitetet i Bergen, Norway University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108425797 DOI: 10.1017/9781108349130 c© Bjørn Ian Dundas 2018 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2018 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalogue record for this publication is available from the British Library. Library of Congress CataloginginPublication Data Names: Dundas, B. I. (Bjorn Ian), author. Title: A short course in differential topology / Bjorn Ian Dundas (Universitetet i Bergen, Norway). Other titles: Cambridge mathematical textbooks. Description: Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2018.  Series: Cambridge mathematical textbooks Identifiers: LCCN 2017061465 ISBN 9781108425797 (hardback ; alk. paper)  ISBN 1108425798 (hardback ; alk. paper) Subjects: LCSH: Differential topology. Classification: LCC QA613.6 .D86 2018  DDC 514/.72–dc23 LC record available at https://lccn.loc.gov/2017061465 ISBN 9781108425797 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or thirdparty internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface page ix 1 Introduction 1 1.1 A Robot’s Arm 1 1.2 The Configuration Space of Two Electrons 6 1.3 State Spaces and Fiber Bundles 7 1.4 Further Examples 9 1.5 Compact Surfaces 11 1.6 Higher Dimensions 16 2 Smooth Manifolds 18 2.1 Topological Manifolds 18 2.2 Smooth Structures 22 2.3 Maximal Atlases 27 2.4 Smooth Maps 32 2.5 Submanifolds 37 2.6 Products and Sums 42 3 The Tangent Space 47 3.1 Germs 50 3.2 Smooth Bump Functions 53 3.3 The Tangent Space 55 3.4 The Cotangent Space 60 3.5 Derivations 68 4 Regular Values 73 4.1 The Rank 73 4.2 The Inverse Function Theorem 76 4.3 The Rank Theorem 78 4.4 Regular Values 83 4.5 Transversality 89 4.6 Sard’s Theorem 92 4.7 Immersions and Imbeddings 93 5 Vector Bundles 98 5.1 Topological Vector Bundles 99 5.2 Transition Functions 105 viii Contents 5.3 Smooth Vector Bundles 108 5.4 Prevector Bundles 111 5.5 The Tangent Bundle 113 5.6 The Cotangent Bundle 120 6 Constructions on Vector Bundles 123 6.1 Subbundles and Restrictions 123 6.2 The Induced Bundle 128 6.3 Whitney Sum of Bundles 131 6.4 Linear Algebra on Bundles 133 6.5 Normal Bundles 139 6.6 Riemannian Metrics 141 6.7 Orientations 145 6.8 The Generalized Gauss Map 146 7 Integrability 148 7.1 Flows and Velocity Fields 148 7.2 Integrability: Compact Case 154 7.3 Local Flows and Integrability 157 7.4 SecondOrder Differential Equations 161 8 Local Phenomena that Go Global 164 8.1 Refinements of Covers 164 8.2 Partition of Unity 166 8.3 Global Properties of Smooth Vector Bundles 170 8.4 An Introduction to Morse Theory 173 8.5 Ehresmann’s Fibration Theorem 182 Appendix A Point Set Topology 191 A.1 Topologies: Open and Closed Sets 192 A.2 Continuous Maps 193 A.3 Bases for Topologies 194 A.4 Separation 195 A.5 Subspaces 195 A.6 Quotient Spaces 196 A.7 Compact Spaces 198 A.8 Product Spaces 199 A.9 Connected Spaces 199 A.10 SetTheoretical Stuff 200 Appendix B Hints or Solutions to the Exercises 202 References 244 Index 246 Preface In his inaugural lecture in 18541, Riemann introduced the concept of an “nfach ausgedehnte Grösse” – roughly something that has “n degrees of freedom” and which we now would call an ndimensional manifold. Examples of manifolds are all around us and arise in many applications, but formulating the ideas in a satisfying way proved to be a challenge inspiring the creation of beautiful mathematics. As a matter of fact, much of the mathematical language of the twentieth century was created with manifolds in mind. Modern texts often leave readers with the feeling that they are getting the answer before they know there is a problem. Taking the historical approach to this didactic problem has several disadvantages. The pioneers were brilliant mathematicians, but still they struggled for decades getting the concepts right. We must accept that we are standing on the shoulders of giants. The only remedy I see is to give carefully chosen examples to guide the mind to ponder over the questions that you would actually end up wondering about even after spending a disproportionate amount of time. In this way I hope to encourage readers to appreciate and internalize the solutions when they are offered. These examples should be concrete. On the other end of the scale, proofs should also be considered as examples: they are examples of successful reasoning. “Here is a way of handling such situations!” However, no amount of reading can replace doing, so there should be many opportunities for trying your hand. In this book I have done something almost unheard of: I provide (sometimes quite lengthy) hints for all the exercises. This requires quite a lot of selfdiscipline from the reader: it is very hard not to peek at the solution too early. There are several reasons for including hints. First and foremost, the exercises are meant to be an integral part of class life. The exercises can be assigned to students who present their solutions in problem sessions, in which case the students must internalize their solution, but at the same time should be offered some moral support to lessen the social stress. Secondly, the book was designed for students who – even if eager to learn – are in need of more support with respect to how one can reason about the material. Trying your hand on the problem, getting stuck, taking a peek to see whether you glimpse an idea, trying again . . . and eventually getting a solution that you believe in and which you can discuss in class is way preferable to not having . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 https://en.wikipedia.org/wiki/Bernhard_Riemann x Preface anything to bring to class. A side effect is that this way makes it permissible to let the students develop parts of the text themselves without losing accountability. Lastly, though this was not a motivation for me, providing hints makes the text better suited for selfstudy. Why This Book? The year I followed the manifold course (as a student), we used Spivak [20], and I came to love the “Great American Differential Geometry book”. At the same time, I discovered a little gem by Bröker and Jänich [4] in the library that saved me on some of the occasions when I got totally befuddled. I spent an inordinate amount of time on that class. Truth be told, there are many excellent books on manifolds out there; to name just three, Lee’s book [13] is beautiful; in a macho way so is Kosinski’s [11]; and Milnor’s pearl [15] will take you all the way from zero to framed cobordisms in 50 pages. Why write one more? Cambridge University Press wanted “A Short Introduction” to precede my orig inal title “Differential Topology”. They were right: this is a far less ambitious text than the ones I have mentioned, and was designed for the students who took my classes. As a student I probably could provide a proof for all the theorems, but if someone asked me to check a very basic fact like “Is this map smooth?” I would feel that it was so for “obvious reasons” and hope for the life of me that no one would ask “why?” The book offers a modern framework while not reducing every thing to some sort of magic. This allows us to take a handson approach; we are less inclined to identify objects without being specific about how they should be iden tified, removing some of the anxiety about “variables” and “coordinates changing” this or that way. Spending time on the basics but still aiming at a onesemester course forces some compromises on this fairly short book. Sadly, topics like Sard’s theorem, Stokes’ theorem, differential forms, de Rham cohomology, differential equations, Rieman nian geometry and surfaces, imbedding theory, Ktheory, singularities, foliations and analysis on manifolds are barely touched upon. At the end of the term, I hope that the reader will have internalized the fun damental ideas and will be able to use the basic language and tools with enough confidence to apply them in other fields, and to embark on more ambitious texts. Also, I wanted to prove Ehresmann’s fibration theorem because I think it is cool. How to Start Reading The core curriculum consists of Chapters 2–8. The introduction in Chapter 1 is not strictly necessary for highly motivated readers who cannot wait to get to the theory, but provides some informal examples and discussions meant to put the later mate rial into some perspective. If you are weak on point set topology, you will probably want to read Appendix A in parallel with Chapter 2. You should also be aware Preface xi of the fact that Chapters 4 and 5 are largely independent, and, apart from a few exercises, can be read in any order. Also, at the cost of removing some exercises and examples, the sections on derivations (Section 3.5), orientations (Section 6.7), the generalized Gauss map (Section 6.8), secondorder differential equations (Sec tion 7.4), the exponential map (Section 8.2.7) and Morse theory (Section 8.4) can be removed from the curriculum without disrupting the logical development of ideas. The cotangent space/bundle material (Sections 3.4 and 5.6) can be omitted at the cost of using the dual tangent bundle from Chapter 6 onward. Do the exercises, and only peek(!) at the hints if you really need to. Prerequisites Apart from relying on standard courses in multivariable analysis and linear algebra, this book is designed for readers who have already completed either a course in analysis that covers the basics of metric spaces or a first course in general topology. Most students will feel that their background in linear algebra could have been stronger, but it is to be hoped that seeing it used will increase their appreciation of things beyond Gaussian elimination. Acknowledgments First and foremost, I am indebted to the students and professors who have used the online notes and given me invaluable feedback. Special thanks go to Håvard Berland, Elise Klaveness, Torleif Veen, Karen Sofie Ronæss, Ivan Viola, Samuel Littig, Eirik Berge, Morten Brun and Andreas Leopold Knutsen. I owe much to a couple of anonymous referees (some unsolicited, but all very helpful) for their diligent reading and many constructive comments. They were probably right to insist that the final version should not actively insult the reader even if it means adopting less colorful language. The people at Cambridge University Press have all been very helpful, and I want especially to thank Clare Dennison, Tom Harris and Steven Holt. My debt to the books [8], [11], [12], [13], [15], [14], [20] and in particular [4] should be evident from the text. I am grateful to UiB for allowing me to do several revisions in an inspiring envi ronment (Kistrand, northern Norway), and to the Hausdorff Institute in Bonn and the University of Copenhagen for their hospitality. The frontispiece is an adaption of one of my Tshirts. Thanks to Vår Iren Hjorth Dundas. Notation We let N = {0, 1, 2, . . . }, Z = {. . . ,−1, 0, 1, . . . }, Q, R and C be the sets of natural numbers, integers, rational numbers, real numbers and complex numbers. If X and Y are two sets, X × Y is the set of ordered pairs (x, y) with x an element in X and y an element in Y . If n is a natural number, we let Rn and Cn be the vector spaces of ordered ntuples of real and complex numbers. Occasionally we xii Preface may identify Cn with R2n . If p = (p1, . . . , pn) ∈ Rn , we let p be the norm√ p21 + · · · + p2n . The sphere of dimension n is the subset Sn ⊆ Rn+1 of all p = (p0, . . . , pn) ∈ Rn+1 with p = 1 (so that S0 = {−1, 1} ⊆ R, and S1 can be viewed as all the complex numbers eiθ of unit length). Given functions f : X → Y and g : Y → Z , we write g f for the composite, and g ◦ f only if the notation is cluttered and the ◦ improves readability. The constellation g · f will occur in the situation where f and g are functions with the same source and target, and where multiplication makes sense in the target. If X and Y are topological spaces, a continuous function f : X → Y is simply referred to as a map. 1 Introduction The earth is round. This may at one point have been hard to believe, but we have grown accustomed to it even though our everyday experience is that the earth is (fairly) flat. Still, the most effective way to illustrate it is by means of maps: a globe (Figure 1.1) is a very neat device, but its global(!) character makes it less than practical if you want to represent fine details. This phenomenon is quite common: locally you can represent things by means of “charts”, but the global character can’t be represented by a single chart. You need an entire atlas, and you need to know how the charts are to be assembled, or, even better, the charts overlap so that we know how they all fit together. The mathe matical framework for working with such situations is manifold theory. Before we start off with the details, let us take an informal look at some examples illustrating the basic structure. 1.1 A Robot’s Arm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . To illustrate a few points which will be important later on, we discuss a concrete situation in some detail. The features that appear are special cases of general phe nomena, and the example should provide the reader with some déjà vu experiences later on, when things are somewhat more obscure. Figure 1.1. A globe. Photo by DeAgostini/Getty Images. 2 Introduction y z x Figure 1.2. A A B B Figure 1.3. Consider a robot’s arm. For simplicity, assume that it moves in the plane, and has three joints, with a telescopic middle arm (see Figure 1.2). Call the vector defining the inner arm x , that for the second arm y and that for the third arm z. Assume x  = z = 1 and y ∈ [1, 5]. Then the robot can reach anywhere inside a circle of radius 7. But most of these positions can be reached in several different ways. In order to control the robot optimally, we need to understand the various configurations, and how they relate to each other. As an example, place the robot at the origin and consider all the possible posi tions of the arm that reach the point P = (3, 0) ∈ R2, i.e., look at the set T of all triples (x, y, z) ∈ R2 × R2 × R2 such that x + y + z = (3, 0), x  = z = 1, y ∈ [1, 5]. We see that, under the restriction x  = z = 1, x and z can be chosen arbitrarily, and determine y uniquely. So T is “the same as” the set {(x, z) ∈ R2 × R2  x  = z = 1}. Seemingly, our space T of configurations resides in fourdimensional space R2 × R2 ∼= R4, but that is an illusion – the space is twodimensional and turns out to be a familiar shape. We can parametrize x and z by angles if we remember to identify the angles 0 and 2π . So T is what you get if you con sider the square [0, 2π] × [0, 2π] and identify the edges as in Figure 1.3. See 1.1 A Robot’s Arm 3 Figure 1.4. 0 1 2 3 4 5 6 s 0 1 2 3 4 5 6 t 1 2 3 4 5 Figure 1.5. www.it.brighton.ac.uk/staff/jt40/MapleAnimations/Torus.html for a nice animation of how the plane model gets glued. In other words, the set T of all positions such that the robot reaches P = (3, 0) may be identified with the torus in Figure 1.4. This is also true topologically in the sense that “close configurations” of the robot’s arm correspond to points close to each other on the torus. 1.1.1 Question What would the space S of positions look like if the telescope got stuck at y = 2? Partial answer to the question: since y = (3, 0) − x − z we could try to get an idea of what points of T satisfy y = 2 by means of inspection of the graph of y. Figure 1.5 is an illustration showing y as a function of T given as a graph over [0, 2π] × [0, 2π], and also the plane y = 2. The desired set S should then be the intersection shown in Figure 1.6. It looks a bit weird before we remember that the edges of [0, 2π] × [0, 2π] should be identified. On the torus it looks perfectly fine; and we can see this if we change our perspective a bit. In order to view T we chose [0, 2π]×[0, 2π] with identifications along the boundary. We could just as well have chosen [−π, π] × [−π, π], and then the picture would have looked like Figure 1.7. It does not touch the boundary, 4 Introduction 0 1 2 3 4 5 6 t 0 1 2 3 4 5 6 s Figure 1.6. –3 –3 –2 –2 –1 –1 0 s 0 1 1 2 2 3 3 Figure 1.7. so we do not need to worry about the identifications. As a matter of fact, S is homeomorphic to the circle (homeomorphic means that there is a bijection between S and the circle, and both the function from the circle to S and its inverse are continuous. See Definition A.2.8). 1.1.2 Dependence on the Telescope’s Length Even more is true: we notice that S looks like a smooth and nice curve. This will not happen for all values of y. The exceptions are y = 1, y = 3 and y = 5. The values 1 and 5 correspond to onepoint solutions. When y = 3 we get a picture like Figure 1.8 (the solution really ought to touch the boundary). We will learn to distinguish between such circumstances. They are qualitatively different in many aspects, one of which becomes apparent if we view the exam ple shown in Figure 1.9 with y = 3 with one of the angles varying in [0, 2π] while the other varies in [−π, π]. With this “cross” there is no way our solution space is homeomorphic to the circle. You can give an interpretation of the picture 1.1 A Robot’s Arm 5 –3 –2 –1 0 1 2 3 t –3 –2 –1 0 1 2 3 s Figure 1.8. –3 –2 –1 0 1 2 3 t 0 1 2 3 4 5 6 s Figure 1.9. above: the straight line is the movement you get if you let x = z (like two wheels of equal radius connected by a coupling rod y on an oldfashioned train), whereas the curved line corresponds to x and z rotating in opposite directions (very unhealthy for wheels on a train). Actually, this cross comes from a “saddle point” in the graph of y as a function of T : it is a “critical” value at which all sorts of bad things can happen. 1.1.3 Moral The configuration space T is smooth and nice, and we get different views on it by changing our “coordinates”. By considering a function on T (in our case the length of y) and restricting to the subset of T corresponding to a given value of our function, we get qualitatively different situations according to what values we are looking at. However, away from the “critical values” we get smooth and nice subspaces, see in particular Theorem 4.4.3. 6 Introduction 1.2 The Configuration Space of Two Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Consider the situation where two electrons with the same spin are lonesome in space. To simplify matters, place the origin at the center of mass. The Pauli exclu sion principle dictates that the two electrons cannot be at the same place, so the electrons are somewhere outside the origin diametrically opposite of each other (assume they are point particles). However, you can’t distinguish the two electrons, so the only thing you can tell is what line they are on, and how far they are from the origin (you can’t give a vector v saying that this points at a chosen electron:−v is just as good). Disregarding the information telling you how far the electrons are from each other (which anyhow is just a matter of scale), we get that the space of possible positions may be identified with the space of all lines through the origin in R3. This space is called the (real) projective plane RP2. A line intersects the unit sphere S2 = {p ∈ R3  p = 1} in exactly two (antipodal) points, and so we get that RP2 can be viewed as the sphere S2 but with p ∈ S2 identified with−p. A point in RP2 represented by p ∈ S2 (and −p) is written [p]. The projective plane is obviously a “manifold” (i.e., can be described by means of charts), since a neighborhood around [p] can be identified with a neighbor hood around p ∈ S2 – as long as they are small enough to fit on one hemisphere. However, I cannot draw a picture of it in R3 without cheating. On the other hand, there is a rather concrete representation of this space: it is what you get if you take a Möbius band (Figure 1.10) and a disk (Figure 1.11), and glue them together along their boundary (both the Möbius band and the disk have boundaries a copy of the circle). You are asked to perform this identification in Exercise 1.5.3. 1.2.1 Moral The moral in this subsection is this: configuration spaces are oftentimes manifolds that do not in any natural way live in Euclidean space. From a technical point of view they often are what can be called quotient spaces (although this example was a rather innocent one in this respect). Figure 1.10. A Möbius band: note that its boundary is a circle. 1.3 State Spaces and Fiber Bundles 7 Figure 1.11. A disk: note that its boundary is a circle. 1.3 State Spaces and Fiber Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The following example illustrates a phenomenon often encountered in physics, and a tool of vital importance for many applications. It is also an illustration of a key result which we will work our way towards: Ehresmann’s fibration theorem, 8.5.10 (named after Charles Ehresmann, 1905–1979)1. It is slightly more involved than the previous example, since it points forward to many concepts and results we will discuss more deeply later, so if you find the going a bit rough, I advise you not to worry too much about the details right now, but come back to them when you are ready. 1.3.1 Qbits In quantum computing one often talks about qbits. As opposed to an ordinary bit, which takes either the value 0 or the value 1 (representing “false” and “true” respec tively), a qbit, or quantum bit, is represented by a complex linear combination (“superposition” in the physics parlance) of two states. The two possible states of a bit are then often called 0〉 and 1〉, and so a qbit is represented by the “pure qbit state” α0〉 + β1〉, where α and β are complex numbers and α2 + β2 = 1 (since the total probability is 1, the numbers α2 and β2 are interpreted as the probabilities that a measurement of the qbit will yield 0〉 and 1〉 respectively). Note that the set of pairs (α, β) ∈ C2 satisfying α2 + β2 = 1 is just another description of the sphere S3 ⊆ R4 = C2. In other words, a pure qbit state is a point (α, β) on the sphere S3. However, for various reasons phase changes are not important. A phase change is the result of multiplying (α, β) ∈ S3 by a unitlength complex number. That is, if z = eiθ ∈ S1 ⊆ C, the pure qbit state (zα, zβ) is a phase shift of (α, β), and these should be identified. The state space is what you get when you identify each pure qbit state with the other pure qbit states you get by a phase change. So, what is the relation between the space S3 of pure qbit states and the state space? It turns out that the state space may be identified with the twodimensional sphere S2 (Figure 1.12), and the projection down to state space η : S3 → S2 may then be given by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 https://en.wikipedia.org/wiki/Charles_Ehresmann 8 Introduction Figure 1.12. The state space S2. Piece of qbit space State space U U × S1 Figure 1.13. The pure qbit states represented in a small open neighborhood U in state space form a cylinder U× S1 (dimension reduced by one in the picture). η(α, β) = (α2 − β2, 2αβ̄) ∈ S2 ⊆ R3 = R× C. Note that η(α, β) = η(zα, zβ) if z ∈ S1, and so η does indeed send all the phase shifts of a given pure qbit to the same point in state space, and conversely, any two pure qbits in preimage of a given point in state space are phase shifts of each other. Given a point in state space p ∈ S2, the space of pure qbit states representing p can be identified with S1 ⊆ C: choose a pure qbit state (α, β) representing p, and note that any other pure qbit state representing p is of the form (zα, zβ) for some unique z ∈ S1. So, can a pure qbit be given uniquely by its associated point in the state space and some point on the circle, i.e., is the space of pure qbit states really S2 × S1 (and not S3 as I previously claimed)? Without more work, it is not at all clear how these copies of S1 lying over each point in S2 are to be glued together: how does this “circle’s worth” of pure qbit states change when we vary the position in state space slightly? The answer comes through Ehresmann’s fibration theorem, 8.5.10. It turns out that η : S3 → S2 is a locally trivial fibration, which means that, in a small neigh borhood U around any given point in state space, the space of pure qbit states does look like U × S1. See Figure 1.13. On the other hand, the global structure is different. In fact, η : S3 → S2 is an important mathematical object for many reasons, and is known as the Hopf fibration. 1.4 Further Examples 9 The input to Ehresmann’s theorem comes in two types. First we have some point set information, which in our case is handled by the fact that S3 is “compact” A.7.1. Secondly, there is a condition which sees only the linear approximations, and which in our case boils down to the fact that any “infinitesimal” movement on S2 is the shadow of an “infinitesimal” movement in S3. This is a question which – given the right language – is settled through a quick and concrete calculation of differentials. We’ll be more precise about this later (this is Exercise 8.5.16). 1.3.2 Moral The idea is the important thing: if you want to understand some complicated model through some simplification, it is often so that the complicated model locally (in the simple model) can be built out of the simple model through multiplying with some fixed space. How these local pictures are glued together to give the global picture is another matter, and often requires other tools, for instance from algebraic topology. In the S3 → S2 case, we see that S3 and S2 × S1 cannot be identified since S3 is simply connected (meaning that any closed loop in S3 can be deformed continuously to a point) and S2 × S1 is not. An important class of examples (of which the above is one) of locally trivial fibrations arises from symmetries: if M is some (configuration) space and you have a “group of symmetries” G (e.g., rotations) acting on M , then you can consider the space M/G of points in M where you have identified two points in M if they can be obtained from each other by letting G act (e.g., one is a rotated copy of the other). Under favorable circumstances M/G will be a manifold and the projection M → M/G will be a locally trivial fibration, so that M is built by gluing together spaces of the form U × G, where U varies over the open subsets of M/G. 1.4 Further Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A short bestiary of manifolds available to us at the moment might look like this. ● The surface of the earth, S2, and higherdimensional spheres, see Exam ple 2.1.5. ● Spacetime is a manifold: general relativity views spacetime as a four dimensional “pseudoRiemannian” manifold. According to Einstein its curva ture is determined by the mass distribution. (Whether the largescale structure is flat or not is yet another question. Current measurements sadly seem to be consistent with a flat largescale structure.) ● Configuration spaces in physics (e.g., the robot in Example 1.1, the two elec trons of Example 1.2 or the more abstract considerations at the very end of Section 1.3.2 above). 10 Introduction ● If f : Rn → R is a map and y a real number, then the inverse image f −1(y) = {x ∈ Rn  f (x) = y} is often a manifold. For instance, if f : R2 → R is the norm function f (x) = x , then f −1(1) is the unit circle S1 (c.f. the discussion of submanifolds in Chapter 4). ● The torus (c.f. the robot in Example 1.1). ● “The real projective plane” RP2 = {All lines in R3 through the origin} (see the twoelectron case in Example 1.2, but also Exercise 1.5.3). ● The Klein bottle2 (see Section 1.5). We end this introduction by studying surfaces in a bit more detail (since they are concrete, and this drives home the familiar notion of charts in more exotic sit uations), and also come up with some inadequate words about higherdimensional manifolds in general. 1.4.1 Charts The spacetime manifold brings home the fact that manifolds must be represented intrinsically: the surface of the earth is seen as a sphere “in space”, but there is no space which should naturally harbor the universe, except the universe itself. This opens up the question of how one can determine the shape of the space in which we live. One way of representing the surface of the earth as the twodimensional space it is (not referring to some ambient threedimensional space), is through an atlas. The shape of the earth’s surface is then determined by how each map in the atlas is to be glued to the other maps in order to represent the entire surface. Just like the surface of the earth is covered by maps, the torus in the robot’s arm was viewed through flat representations. In the technical sense of the word, the representation was not a “chart” (see Definition 2.1.1) since some points were covered twice (just as Siberia and Alaska have a tendency to show up twice on some European maps). It is allowed to have many charts covering Fairbanks in our atlas, but on each single chart it should show up at most once. We may fix this problem at the cost of having to use more overlapping charts. Also, in the robot example (as well as the twoelectron and qbit examples) we saw that it was advantageous to operate with more charts. Example 1.4.2 To drive home this point, please play Jeff Weeks’ “Torus Games” on www.geometrygames.org/TorusGames/ for a while. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 wwwgroups.dcs.stand.ac.uk/∼history/Biographies/Klein.html 1.5 Compact Surfaces 11 1.5 Compact Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . This section is rather autonomous, and may be read at leisure at a later stage to fill in the intuition on manifolds. 1.5.1 The Klein Bottle To simplify we could imagine that we were twodimensional beings living in a static closed surface. The sphere and the torus are familiar surfaces, but there are many more. If you did Example 1.4.2, you were exposed to another surface, namely the Klein bottle. This has a plane representation very similar to that of the torus: just reverse the orientation of a single edge (Figure 1.14). Although the Klein bottle is an easy surface to describe (but frustrating to play chess on), it is too complicated to fit inside our threedimensional space (again a manifold is not a space inside a flat space, it is a locally Euclidean space). The best we can do is to give an “immersed” (i.e., allowing selfintersections) picture (Figure 1.15). Speaking of pictures: the Klein bottle makes a surprising entré in image anal ysis. When analyzing the ninedimensional space of 3 × 3 patches of grayscale pixels, it is of importance – for instance if you want to implement some compres sion technique – to know what highcontrast configurations occur most commonly. Carlsson, Ishkhanov, de Silva and Zomorodian show in [5] that the subspace of “most common highcontrast pixel configurations” actually “is” a Klein bottle. a a b b Figure 1.14. A plane representation of the Klein bottle: identify along the edges in the direction indicated. Figure 1.15. A picture of the Klein bottle forced into our threedimensional space: it is really just a shadow since it has selfintersections. If you insist on putting this twodimensional manifold into a flat space, you must have at least four dimensions available. 12 Introduction Their results have been used to develop a compression algorithm based on a “Klein bottle dictionary”. 1.5.2 Classification of Compact Surfaces As a matter of fact, it turns out that we can write down a list of all compact surfaces (compact is defined in Appendix A, but informally should be thought of as “closed and of bounded size”). First of all, surfaces may be divided into those that are orientable and those that are not. Orientable means that there are no loops by which twodimensional beings living in the surface can travel and return home as their mirror images. (Is the universe nonorientable? Is that why some people are left handed?) All connected compact orientable surfaces can be obtained by attaching a finite number of handles to a sphere. The number of handles attached is referred to as the genus of the surface. A handle is a torus with a small disk removed (see Figure 1.16). Note that the boundary of the hole on the sphere and the boundary of the hole on each handle are all circles, so we glue the surfaces together in a smooth manner along their common boundary (the result of such a gluing process is called the connected sum, and some care is required). Thus all orientable compact surfaces are surfaces of pretzels with many holes (Figure 1.17). There are nonorientable surfaces too (e.g., the Klein bottle). To make them, con sider a Möbius band3 (Figure 1.18). Its boundary is a circle, so after cutting a hole in a surface you may glue in a Möbius band. If you do this on a sphere you get the projective plane (this is Exercise 1.5.3). If you do it twice you get the Klein bottle. Any nonorientable compact surface can be obtained by cutting Figure 1.16. A handle: ready to be attached to another 2manifold with a small disk removed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 wwwgroups.dcs.stand.ac.uk/∼history/Biographies/Mobius.html 1.5 Compact Surfaces 13 Figure 1.17. An orientable surface of genus g is obtained by gluing g handles (the smoothing out has yet to be performed in these pictures). Figure 1.18. A Möbius band: note that its boundary is a circle. a finite number of holes in a sphere and gluing in the corresponding number of Möbius bands. The reader might wonder what happens if we mix handles and Möbius bands, and it is a strange fact that if you glue g handles and h > 0 Möbius bands you get the same as if you had glued h + 2g Möbius bands! For instance, the projective plane with a handle attached is the same as the Klein bottle with a Möbius band glued onto it. But fortunately this is it; there are no more identifications among the surfaces. So, any (connected compact) surface can be obtained by cutting g holes in S2 and either gluing in g handles or gluing in g Möbius bands. For a detailed discussion the reader may turn to Chapter 9 of Hirsch’s book [8]. 1.5.3 Plane Models If you find such descriptions elusive, you may derive some comfort from the fact that all compact surfaces can be described similarly to the way we described the torus. If we cut a hole in the torus we get a handle. This may be represented by plane models as in Figure 1.19: identify the edges as indicated. If you want more handles you just glue many of these together, so that a gholed torus can be represented by a 4ggon where two and two edges are iden tified. (See Figure 1.20 for the case g = 2; the general case is similar. See 14 Introduction a a b b the boundary a a b b Figure 1.19. Two versions of a plane model for the handle: identify the edges as indicated to get a torus with a hole in. a a′ a a′ b b b′ b′ Figure 1.20. A plane model of the orientable surface of genus two. Glue corresponding edges together. The dotted line splits the surface up into two handles. a a the boundary Figure 1.21. A plane model for the Möbius band: identify the edges as indicated. When gluing it onto something else, use the boundary. also www.rogmann.org/math/tori/torus2en.html for instruction on how to sew your own two and threeholed torus.) It is important to have in mind that the points on the edges in the plane models are in no way special: if we change our point of view slightly we can get them to be in the interior. We have plane models for gluing in Möbius bands too (see Figure 1.21). So a sur face obtained by gluing h Möbius bands to h holes on a sphere can be represented by a 2hgon, with pairwise identification of edges. Example 1.5.1 If you glue two plane models of the Möbius band along their boundaries you get the picture in Figure 1.22. This represents the Klein bottle, but it is not exactly the same plane representation as the one we used earlier (Figure 1.14). To see that the two plane models give the same surface, cut along the line c in the diagram on the left in Figure 1.23. Then take the two copies of the line a and glue them together in accordance with their orientations (this requires that you flip one 1.5 Compact Surfaces 15 a a a′a′ Figure 1.22. Gluing two flat Möbius bands together. The dotted line marks where the bands were glued together. a a a′a′ c a a′ a′ c c Figure 1.23. Cutting along c shows that two Möbius bands glued together amount to the Klein bottle. of your triangles). The resulting diagram, which is shown to the right, is (a rotated and slanted version of) the plane model we used before for the Klein bottle. Exercise 1.5.2 Prove by a direct cutandpaste argument that what you get by adding a handle to the projective plane is the same as what you get if you add a Möbius band to the Klein bottle. Exercise 1.5.3 Prove that the real projective plane RP2 = {All lines in R3 through the origin} is the same as what you get by gluing a Möbius band to a sphere. Exercise 1.5.4 See whether you can find out what the “Euler number”4 (or Euler characteristic)5 is. Then calculate it for various surfaces using the plane models. Can you see that both the torus and the Klein bottle have Euler number zero? The sphere has Euler number 2 (which leads to the famous theorem V − E + F = 2 for all surfaces bounding a “ball”) and the projective plane has Euler number 1. The surface of Exercise 1.5.2 has Euler number −1. In general, adding a handle reduces the Euler number by two, and adding a Möbius band reduces it by one. Exercise 1.5.5 If you did Exercise 1.5.4, design an (immensely expensive) experiment that could be performed by twodimensional beings living in a compact orientable surface, determining the shape of their universe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 wwwgroups.dcs.stand.ac.uk/∼history/Biographies/Euler.html 5 http://en.wikipedia.org/wiki/Euler_characteristic 16 Introduction 1.6 Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Although surfaces are fun and concrete, next to no reallife applications are two or threedimensional. Usually there are zillions of variables at play, and so our manifolds will be correspondingly complex. This means that we can’t continue to be vague (the previous sections indicated that even in three dimensions things become complicated). We need strict definitions to keep track of all the structure. However, let it be mentioned at the informal level that we must not expect to have such a nice list of higherdimensional manifolds as we had for compact surfaces. Classification problems for higherdimensional manifolds constitute an extremely complex and interesting business we will not have occasion to delve into. It opens new fields of research using methods both from algebra and from analysis that go far beyond the ambitions of this text. 1.6.1 The Poincaré Conjecture and Thurston’s Geometrization Conjecture In 1904 H. Poincaré 6 conjectured that any simply connected compact and closed 3 manifold is homeomorphic to the 3sphere. This problem remained open for almost 100 years, although the corresponding problem was resolved in higher dimensions by S. Smale7 (1961; for dimensions greater than 4, see [18]) and M. Freedman8 (1982; in dimension 4, see [7]). In the academic year 2002/2003 G. Perelman9 published a series of papers build ing on previous work by R. Hamilton10, which by now have come to be widely regarded as the core of a proof of the Poincaré conjecture. The proof relies on an analysis of the “Ricci flow” deforming the curvature of a manifold in a man ner somehow analogous to the heat equation, smoothing out irregularities. Our encounter with flows will be much more elementary, but will still prove essential in the proof of Ehresmann’s fibration theorem, 8.5.10. Perelman was offered the Fields Medal for his work in 2006, but spectacularly refused it. In this way he created much more publicity for the problem, mathemat ics and himself than would have otherwise been thinkable. In 2010 Perelman was also awarded the USD1M Millennium Prize from the Clay Mathematics Institute11. Again he turned down the prize, saying that Hamilton’s contribution in proving the Poincaré conjecture was “no less than mine” (see, e.g., the Wikipedia entry12 on the Poincaré conjecture for an updated account). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 wwwgroups.dcs.stand.ac.uk/∼history/Biographies/Poincare.html 7 wwwgroups.dcs.stand.ac.uk/∼history/Biographies/Smale.html 8 wwwhistory.mcs.standrews.ac.uk/Mathematicians/Freedman.html 9 http://en.wikipedia.org/wiki/Grigori_Perelman 10 http://en.wikipedia.org/wiki/Richard_Hamilton 11 www.claymath.org 12 http://en.wikipedia.org/wiki/Poincare_conjecture 1.6 Higher Dimensions 17 Of far greater consequence is Thurston’s geometrization conjecture. This conjec ture was proposed by W. Thurston13 in 1982. Any 3manifold can be decomposed into prime manifolds, and the conjecture says that any prime manifold can be cut along tori, so that the interior of each of the resulting manifolds has one of eight geometric structures with finite volume. See, e.g., the Wikipedia page14 for fur ther discussion and references to manuscripts with details of the proof filling in Perelman’s sketch. 1.6.2 The History of Manifolds Although it is a fairly young branch of mathematics, the history behind the the ory of manifolds is rich and fascinating. The reader should take the opportunity to check out some of the biographies at The MacTutor History of Mathematics archive15 or the Wikipedia entries of the mathematicians mentioned by name in the text (I have occasionally provided direct links). There is also a page called History Topics: Geometry and Topology Index16 which is worthwhile spending some time with. Of printed books, I have found Jean Dieudonné’s book [6] especially helpful (although it is mainly concerned with topics beyond the scope of this book). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 wwwgroups.dcs.stand.ac.uk/∼history/Biographies/Thurston.html 14 http://en.wikipedia.org/wiki/Geometrization_conjecture 15 wwwgroups.dcs.stand.ac.uk/∼history/index.html 16 wwwgroups.dcs.stand.ac.uk/∼history/Indexes/Geometry_Topology.html 2 Smooth Manifolds 2.1 Topological Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Let us get straight to our object of study. The terms used in the definition are explained immediately below the box. If words like “open” and “topology” are new to you, you are advised to read Appendix A on pointset topology in parallel with this chapter. Definition 2.1.1 An ndimensional topological manifold M is a Hausdorff topological space with a countable basis for the topology which is locally homeomorphic to Rn . The last point (locally homeomorphic to Rn – implicitly with the metric topology – also known as Euclidean space, see Definition A.1.8) means that for every point p ∈ M there is an open neighborhood U of p in M , an open set U ′ ⊆ Rn and a homeomorphism (Definition A.2.5) x : U → U ′. We call such an x : U → U ′ a chart and U a chart domain (Figure 2.1). U U′ X M Figure 2.1. 2.1 Topological Manifolds 19 A collection of charts {xα : Uα → U ′α} covering M (i.e., such that the union⋃ Uα of the chart domains is M) is called an atlas. Note 2.1.2 The conditions that M should be “Hausdorff” (Definition A.4.1) and have a “countable basis for its topology” (Section A.3) will not play an important rôle for us for quite a while. It is tempting to just skip these conditions, and come back to them later when they actually are important. As a matter of fact, on a first reading I suggest you actually do this. Rest assured that all subsets of Euclidean spaces satisfy these conditions (see Corollary A.5.6). The conditions are there in order to exclude some pathological creatures that are locally homeomorphic to Rn , but are so weird that we do not want to consider them. We include the conditions at once so as not to need to change our definition in the course of the book, and also to conform with usual language. Note that Rn itself is a smooth manifold. In particular, it has a countable basis for its topology (c.f. Exercise A.3.4). The requirement that there should be a countable basis for the topology could be replaced by demanding the existence of a countable atlas. Note 2.1.3 When saying that “M is a manifold” without specifying its dimension, one could envision that the dimension need not be the same everywhere. We only really care about manifolds of a fixed dimension, and even when allowing the dimension to vary, each connected component has a unique dimension. Conse quently, you may find that we’ll not worry about this and in the middle of an argument say something like “let n be the dimension of M” (and proceed to talk about things that concern only one component at a time). Example 2.1.4 Let U ⊆ Rn be an open subset. Then U is an nmanifold. Its atlas needs only one chart, namely the identity map id : U = U . As a subexample we have the open ndisk En = {p ∈ Rn  p < 1}. The notation En has its disadvantages. You may find it referred to as Bn , Bn(1), En(1), B1(0), N R n 1 (0) . . . in other texts. Example 2.1.5 The nsphere Sn = {p ∈ Rn+1  p = 1} is an ndimensional manifold. To see that Sn is locally homeomorphic to Rn we may proceed as follows. Write a point in Rn+1 as an n + 1 tuple indexed from 0 to n: p = (p0, p1, . . . , pn). To give an atlas for Sn , consider the open sets U k,0 = {p ∈ Sn  pk > 0}, U k,1 = {p ∈ Sn  pk < 0} 20 Smooth Manifolds U 0,1U 0,0 U 1,0 U 1,1 Figure 2.2. U1,0 D1 Figure 2.3. for k = 0, . . . , n, and let xk,i : U k,i → En be the projection to the open ndisk En given by deleting the kth coordinate: (p0, . . . , pn) → (p0, . . . , p̂k, . . . , pn) = (p0, . . . , pk−1, pk+1, . . . , pn) (the “hat” in p̂k is a common way to indicate that this coordinate should be deleted). See Figures 2.2 and 2.3. (The nsphere is Hausdorff and has a countable basis for its topology by Corollary A.5.6 simply because it is a subspace of Rn+1.) Exercise 2.1.6 Check that the proposed charts xk,i for Sn in the previous example really are homeomorphisms. Exercise 2.1.7 We shall later see that an atlas with two charts suffices on the sphere. Why is there no atlas for Sn with only one chart? Example 2.1.8 The real projective nspace RPn is the set of all straight lines through the origin in Rn+1. As a topological space, it is the quotient space (see Section A.6) 2.1 Topological Manifolds 21 RPn = (Rn+1 \ {0})/∼, where the equivalence relation is given by p ∼ q if there is a nonzero real number λ such that p = λq. Since each line through the origin intersects the unit sphere in two (antipodal) points, RPn can alternatively be described as Sn/∼, where the equivalence relation is p ∼ −p. The real projective nspace is an n dimensional manifold, as we shall see below. If p = (p0, . . . , pn) ∈ Rn+1 \ {0} we write [p] for its equivalence class considered as a point in RPn . For 0 ≤ k ≤ n, let U k = {[p] ∈ RPnpk �= 0}. Varying k, this gives an open cover of RPn (why is U k open in RPn?). Note that the projection Sn → RPn when restricted to U k,0 ∪U k,1 = {p ∈ Snpk �= 0} gives a twotoone correspondence between U k,0 ∪U k,1 and U k . In fact, when restricted to U k,0 the projection Sn → RPn yields a homeomorphism U k,0 ∼= U k . The homeomorphism U k,0 ∼= U k together with the homeomorphism xk,0 : U k,0 → En = {p ∈ Rn  p < 1} of Example 2.1.5 gives a chart U k → En (the explicit formula is given by sending [p] ∈ U k to (pk /(pk p)) (p0, . . . , p̂k, . . . , pn). Letting k vary, we get an atlas for RPn . We can simplify this somewhat: the following atlas will be referred to as the standard atlas for RPn . Let xk : U k → Rn [p] → 1 pk (p0, . . . , p̂k, . . . , pn) . Note that this is well defined since (1/pk)(p0, . . . , p̂k, . . . , pn) = (1/(λpk))(λp0, . . . , λ̂pk, . . . , λpn). Furthermore, xk is a bijective function with inverse given by( xk )−1 (p0, . . . , p̂k, . . . , pn) = [p0, . . . , 1, . . . , pn] (note the convenient cheating in indexing the points in Rn). In fact, xk is a homeomorphism: xk is continuous since the composite U k,0 ∼= U k → Rn is; and (xk)−1 is continuous since it is the composite Rn → {p ∈ Rn+1  pk �= 0} → U k , where the first map is given by (p0, . . . , p̂k, . . . , pn) → (p0, . . . , 1, . . . , pn) and the second is the projection. (That RPn is Hausdorff and has a countable basis for its topology is shown in Exercise A.7.5.) 22 Smooth Manifolds Note 2.1.9 It is not obvious at this point that RPn can be realized as a subspace of a Euclidean space (we will show it can in Theorem 8.2.6). Note 2.1.10 We will try to be consistent in letting the charts have names like x and y. This is sound practice since it reminds us that what charts are good for is to give “local coordinates” on our manifold: a point p ∈ M corresponds to a point x(p) = (x1(p), . . . , xn(p)) ∈ Rn. The general philosophy when studying manifolds is to refer back to properties of Euclidean space by means of charts. In this manner a successful theory is built up: whenever a definition is needed, we take the Euclidean version and require that the corresponding property for manifolds is the one you get by saying that it must hold true in “local coordinates”. Example 2.1.11 As we defined it, a topological manifold is a topological space with certain properties. We could have gone about this differently, minimizing the rôle of the space at the expense of talking more about the atlas. For instance, given a set M , a collection {Uα}α∈A of subsets of M such that⋃ α∈A Uα = M (we say that {Uα}α∈A covers M) and a collection of injections (one toone functions) {xα : Uα → Rn}α∈A, assume that if α, β ∈ A then the bijection xα(Uα∩Uβ)→ xβ(Uα∩Uβ) sending q to xβxα−1(q) is a continuous map between open subsets of Rn . The declaration that U ⊂ M is open if for all α ∈ A we have that xα(U ∩Uα) ⊆ Rn is open determines a topology on M . If this topology is Hausdorff and has a countable basis for its topology, then M is a topological manifold. This can be achieved if, for instance, we have that (1) for p, q ∈ M , either there is an α ∈ A such that p, q ∈ Uα or there are α, β ∈ A such that Uα and Uβ are disjoint with p ∈ Uα and q ∈ Uβ and (2) there is a countable subset B ⊆ A such that ⋃β∈B Uβ = M . 2.2 Smooth Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . We will have to wait until Definition 2.3.5 for the official definition of a smooth manifold. The idea is simple enough: in order to do differential topology we need that the charts of the manifolds are glued smoothly together, so that our questions regarding differentials or the like do not get different answers when interpreted through different charts. Again “smoothly” must be borrowed from the Euclidean world. We proceed to make this precise. Let M be a topological manifold, and let x1 : U1 → U ′1 and x2 : U2 → U ′2 be two charts on M with U ′1 and U ′ 2 open subsets of R n . Assume that U12 = U1 ∩U2 is nonempty. 2.2 Smooth Structures 23 Figure 2.4. Then we may define a chart transformation as shown in Figure 2.4 (sometimes called a “transition map”) x12 : x1(U12)→ x2(U12) by sending q ∈ x1(U12) to x12(q) = x2x−11 (q) (in function notation we get that x12 = ( x2U12 ) ◦ (x1U12)−1 : x1(U12)→ x2(U12), where we recall that “U12” means simply “restrict the domain of definition to U12”). The picture of the chart transformation above will usually be recorded more succinctly as U12 x1U12 ����� �� �� �� x2U12 ��� �� �� �� �� x1(U12) x2(U12). This makes things easier to remember than the occasionally awkward formulae. The restrictions, like in x1U12 , clutter up the notation, and if we’re pretty sure no confusion can arise we may in the future find ourselves writing variants like x2x −1 1 U12 or even x2x−11 when we should have written (x2U12)(x1U12)−1. This is common practice, but in the beginning you should try to keep everything in place and relax your notation only once you are sure what you actually mean. Definition 2.2.1 A map f between open subsets of Euclidean spaces is said to be smooth if all the higherorder partial derivatives exist and are continuous. A smooth map f between open subsets of Rn is said to be a diffeomorphism if it has a smooth inverse f −1. 24 Smooth Manifolds The chart transformation x12 is a function from an open subset of Rn to another, and it makes sense to ask whether it is smooth or not. Definition 2.2.2 An atlas on a manifold is smooth (or C∞) if all the chart transformations are smooth. Note 2.2.3 Note that, if x12 is a chart transformation associated with a pair of charts in an atlas, then x−112 is also a chart transformation. Hence, saying that an atlas is smooth is the same as saying that all the chart transformations are diffeomorphisms. Note 2.2.4 We are interested only in the infinitely differentiable case, but in some situations it is sensible to ask for less. For instance, we could require that all chart transformations are C1 (all the single partial differentials exist and are continuous). For a further discussion, see Note 2.3.7 below. One could also ask for more, for instance that all chart transformations are ana lytic functions – giving the notion of an analytic manifold. However, the difference between smooth and analytic is substantial, as can be seen from Exercise 2.2.14. Example 2.2.5 Let U ⊆ Rn be an open subset. Then the atlas whose only chart is the identity id : U = U is smooth. Example 2.2.6 The atlas U = {(xk,i ,U k,i )0 ≤ k ≤ n, 0 ≤ i ≤ 1} we gave on the nsphere Sn in Example 2.1.5 is a smooth atlas. To see this, look at the example U = U 0,0 ∩ U 1,1 shown in Figure 2.5 and consider the associated chart transformation( x1,1U ) ◦ (x0,0U )−1 : x0,0(U )→ x1,1(U ). P U0,0 ⊃ U1,1 Figure 2.5. How the point p in x0,0(U) is mapped to x1,1(x0,0)−1(p). 2.2 Smooth Structures 25 First we calculate the inverse of x0,0: Let p = (p1, . . . , pn) be a point in the open disk En , then ( x0,0 )−1 (p) = (√ 1− p2, p1, . . . , pn ) (we choose the positive square root, since we consider x0,0). Furthermore, x0,0(U ) = {(p1, . . . , pn) ∈ Enp1 < 0}. Finally we get that if p ∈ x0,0(U ) then x1,1 ( x0,0 )−1 (p) = (√ 1− p2, p̂1, p2, . . . , pn ) . This is a smooth map, and on generalizing to other indices we get that we have a smooth atlas for Sn . Example 2.2.7 There is another useful smooth atlas on Sn , given by stereo graphic projection. This atlas has only two charts, (x+,U+) and (x−,U−). The chart domains are U+ = {p ∈ Sn  p0 > −1}, U− = {p ∈ Sn  p0 < 1}, and x+ : U+ → Rn is given by sending a point p in Sn to the intersection x+(p) of the (“hyper”) plane Rn = {(0, p1, . . . , pn) ∈ Rn+1} and the straight line through the South pole S = (−1, 0, . . . , 0) and p (see Figure 2.6). Similarly for x−, using the North pole instead. Note that both x+ and x− are homeomorphisms onto all of Rn S p x+(p) x−(p) (p1, . . . , pn) (p1, . . . , pn) p0 p p0 N Figure 2.6. 26 Smooth Manifolds To check that there are no unpleasant surprises, one should write down the formulae: x+(p) = 1 1+ p0 (p1, . . . , pn), x−(p) = 1 1− p0 (p1, . . . , pn). We observe that this defines homeomorphisms U± ∼= Rn . We need to check that the chart transformations are smooth. Consider the chart transformation x+ ( x− )−1 defined on x−(U− ∩U+) = Rn \ {0}. A small calculation gives that if q ∈ Rn then ( x− )−1 (q) = 1 1+ q2 (q 2 − 1, 2q) (solve the equation x−(p) = q with respect to p), and so x+ ( x− )−1 (q) = 1q2 q, which is smooth. A similar calculation for the other chart transformation yields that {(x+,U+), (x−,U−)} is a smooth atlas. Exercise 2.2.8 Verify that the claims and formulae in the stereographic projection example are correct. Note 2.2.9 The last two examples may be somewhat worrisome: the sphere is the sphere and these two atlases are two manifestations of the “same” sphere, are they not? We address questions of this kind in the next chapter, such as “When do two different atlases describe the same smooth manifold?” You should, however, be aware that there are “exotic” smooth structures on spheres, i.e., smooth atlases on the topological manifold Sn which describe smooth structures essentially different from the one(s?) we have described (but only in high dimensions). See in particu lar Exercise 2.3.10 and Note 2.4.13. Furthermore, there are topological manifolds which cannot be given smooth structures. Example 2.2.10 The atlas we gave the real projective space is smooth. As an example consider the chart transformation x2(x0)−1: if p2 �= 0 then x2 ( x0 )−1 (p1, . . . , pn) = 1 p2 (1, p1, p3, . . . , pn). 2.3 Maximal Atlases 27 Exercise 2.2.11 Show in all detail that the complex projective nspace CPn = (Cn+1 \ {0})/∼, where z ∼ w if there exists a λ ∈ C \ {0} such that z = λw, is a compact 2n dimensional manifold. If your topology is not that strong yet, focus on the charts and chart transformations. Exercise 2.2.12 There is a convenient smooth atlas for the circle S1, whose charts we will refer to as angle charts. For each θ0 ∈ R consider the homeomorphism (θ0, θ0 + 2π)→ S1 − {eiθ0} given by sending θ to eiθ . Call the inverse xθ0 . Check that {(xθ0, S1 − eiθ0)}θ0 is a smooth atlas. Exercise 2.2.13 Give the boundary of the square the structure of a smooth manifold. Exercise 2.2.14 Let λ : R→ R be defined by λ(t) = { 0 for t ≤ 0 e−1/t for t > 0. This is a smooth function with values between zero and one. Note that all deriva tives at zero are zero: the McLaurin series fails miserably and λ is definitely not analytic. 2.3 Maximal Atlases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . We easily see that some manifolds can be equipped with many different smooth atlases. An example is the circle. Stereographic projection gives a different atlas than what you get if you for instance parametrize by means of the angle (Exam ple 2.2.7 vs. Exercise 2.2.12). But we do not want to distinguish between these two “smooth structures”, and in order to systematize this we introduce the concept of a maximal atlas. Definition 2.3.1 Let M be a manifold and A a smooth atlas on M . Then we define D(A) as the following set of charts on M : D(A) = ⎧⎨⎩charts y : V → V ′ on M ∣∣∣∣∣∣ for all charts (x,U ) in A, the composite x W (yW )−1 : y(W )→ x(W ) is a diffeomorphism, where W = U ∩ V ⎫⎬⎭ . Lemma 2.3.2 Let M be a manifold and A a smooth atlas on M. Then D(A) is a smooth atlas. 28 Smooth Manifolds Proof. Let y : V → V ′ and z : W → W ′ be two charts in D(A). We have to show that zV∩W ◦ (yV∩W )−1 is smooth. Let q be any point in y(V ∩ W ). We prove that z ◦ y−1 is smooth in a neighborhood of q. Choose a chart x : U → U ′ in A with y−1(q) ∈ U . Letting O = U ∩ V ∩W , we get that zO ◦ (yO)−1 = zO ◦ ((x O)−1 ◦ x O) ◦ (yO)−1 = (zO ◦ (x O)−1) ◦ (x O ◦ (yO)−1)) . Since y and z are in D(A) and x is in A we have by definition that both the maps in the composite above are smooth, and we are done. � The crucial equation can be visualized by the following diagram: O yO ���� �� �� �� � x O �� zO ��� �� �� �� �� y(O) x(O) z(O). Going up and down with x O in the middle leaves everything fixed so the two functions from y(O) to z(O) are equal. Definition 2.3.3 A smooth atlas is maximal if there is no strictly bigger smooth atlas containing it. Exercise 2.3.4 Given a smooth atlas A, prove that D(A) is maximal. Hence any smooth atlas is a subset of a unique maximal smooth atlas. Definition 2.3.5 A smooth structure on a topological manifold is a maximal smooth atlas. A smooth manifold (M,A) is a topological manifold M equipped with a smooth structure A. A differentiable manifold is a topological manifold for which there exists a smooth structure. V ′ = y (V ) • q y ( )U ⊃ V ⊃ W y (V ⊃ W) Figure 2.7. 2.3 Maximal Atlases 29 Note 2.3.6 The following terms are synonymous: smooth, differential and C∞. Note 2.3.7 We are interested only in the smooth case, but in some situations it is sensible to ask for less. For instance, we could require that all chart transformations are C1 (all the single partial differentials exist and are continuous). However, the distinction is not really important since having an atlas with C1 chart transforma tions implies that there is a unique maximal smooth atlas such that the mixed chart transformations are C1 (see, e.g., Theorem 2.9 in Chapter 2 of [8]). Note 2.3.8 In practice we do not give the maximal atlas, but choose only a small practical smooth atlas and apply D to it. Often we write just M instead of (M,A) if A is clear from the context. Exercise 2.3.9 To check that two smooth atlases A and B give the same smooth structure on M (i.e., that D(A) = D(B)) it suffices to verify that for each p ∈ M there are charts (x,U ) ∈ A) and (y, V ) ∈ B with p ∈ W = U ∩ V such that x W (yW )−1 : y(W )→ x(W ) is a diffeomorphism. Exercise 2.3.10 Show that the two smooth atlases we have defined on S n (the standard atlas in Example 2.1.5 and the stereographic projections of Example 2.2.7) are contained in a common maximal atlas. Hence they define the same smooth manifold, which we will simply call the (standard smooth) sphere. Exercise 2.3.11 Choose your favorite diffeomorphism x : Rn → Rn . Why is the smooth structure generated by x equal to the smooth structure generated by the identity? What does the maximal atlas for this smooth structure (the only one we’ll ever consider) on Rn look like? Exercise 2.3.12 Prove that any smooth manifold (M,A) has a countable smooth atlas V (so that D(V) = A). Exercise 2.3.13 Prove that the atlas given by the angle charts in Exercise 2.2.12 gives the standard smooth structure on S1. Following up Example 2.1.11 we see that we can construct smooth manifolds from scratch, without worrying too much about the topology. Lemma 2.3.14 Given (1) a set M, (2) a collection A of subsets of M, and (3) an injection xU : U → Rn for each U ∈ A, such that 30 Smooth Manifolds (1) there is a countable subcollection of A which covers M, (2) for p, q ∈ M, either there is a U ∈ A such that p, q ∈ U or there are U, V ∈ A such that U and V are disjoint with p ∈ U and q ∈ V , and (3) if U, V ∈ A then the bijection xU (U ∩ V ) → xV (U ∩ V ) sending q to xV xU−1(q) is a smooth map between open subsets of Rn, then there is a unique topology on M such that (M,D({(xU ,U )}U∈A)) is a smooth manifold. Proof. For the xU s to be homeomorphisms we must have that a subset W ⊆ M is open if and only if for all U ∈ A the set xU (U ∩ W ) is an open subset of Rn . As before, M is a topological manifold, and by the last condition {(xU ,U )}U∈A is a smooth atlas. � Example 2.3.15 As an example of how Lemma 2.3.14 can be used to construct smooth manifolds, we define a family of very important smooth manifolds called the Grassmann manifolds (after Hermann Grassmann (1809–1877))1. These mani folds show up in a number of applications, and are important to the theory of vector bundles (see for instance Section 6.8). For 0 < k ≤ n, let Gr(k,Rn) (the notation varies in the literature) be the set of all kdimensional linear subspaces of Rn . Note that Gr(1,Rn+1) is nothing but the projective space RPn . We will equip Gr(k,Rn) with the structure of an (n − k)kdimensional smooth manifold, the Grassmann manifold. Doing this properly requires some care, but it is worth your while in that it serves the dual purpose of making the structure of this important space clearer as well as driving home some messages about projections that your linear algebra course might have been too preoccupied with multiplying matrices to make apparent. If V,W ⊆ Rn are linear subspaces, we let prV : Rn → V be the orthogonal projection to V (with the usual inner product) and prVW : W → V the restriction of prV to W . We let Hom(V,W ) be the vector space of all linear maps from V to W . Con cretely, and for the sake of the smoothness arguments below, using the standard basis for Rn we may identify Hom(V,W ) with the dim(V ) · dim(W )dimensional linear subspace of the space of n × n matrices A with the property that, if v ∈ V and v′ ∈ V⊥, then Av ∈ W and Av′ = 0. (Check that the map {A ∈ Mn(R)  v ∈ V, v′ ∈V⊥ ⇒ Av ∈ W, Av′ = 0} → Hom(V,W ) A → {v → Av} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 https://en.wikipedia.org/wiki/Hermann_Grassmann 2.3 Maximal Atlases 31 is an isomorphism with inverse given by representing a linear transformation as a matrix.) If V ∈ Gr(k,Rn), consider the set UV = {W ∈ Gr(k,Rn) W ∩ V⊥ = 0}. We will give “charts” of the form xV : UV → Hom(V, V⊥), to which the reader might object that Hom(V, V⊥) is not as such an open subset of Euclidean space. This is not of the essence, because Hom(V, V⊥) is isomorphic to the vector space M(n−k)k(R) of all (n− k)× k matrices (choose bases V ∼= Rk and V⊥ ∼= Rn−k), which again is isomorphic to R(n−k)k . Another characterization of UV is as the set of all W ∈ Gr(k,Rn) such that prVW : W → V is an isomorphism. Let xV : UV → Hom(V, V⊥) send W ∈ UV to the composite xV (W ) : V (pr V W ) −1 −−−−→ W pr V⊥ W−−−→ V⊥. See Figure 2.8. Varying V we get a smooth atlas {(xV ,UV )}V∈Gr(k,Rn) for Gr(k,Rn). Exercise 2.3.16 Prove that the proposed structure on the Grassmann manifold Gr(k,R n) in Exam ple 2.3.15 actually is a smooth atlas which endows Gr(k,Rn) with the structure of a smooth manifold. Note 2.3.17 In Chapter 1 we used the word “boundary” for familiar objects like the closed unit disk (whose boundary is the unit circle). Generally, the notion of a smooth ndimensional manifold with boundary M is defined exactly as we defined a smooth ndimensional manifold, except that the charts x : U → U ′ in our atlas are required to be homeomorphisms to open subsets U ′ of the half space Hn = {(p1, . . . , pn) ∈ Rn  p1 ≥ 0} V ⊥ W V υ xv(W) (υ) Figure 2.8. 32 Smooth Manifolds (i.e., U ′ = Hn ∩ Ũ ′, where Ũ ′ is an open subset of Rn). If y : V → V ′ is another chart in the atlas, the chart transformation y ◦ x−1x(U∩V ) is “smooth”, but what do we mean by a smooth map f : V ′ → W ′ when V ′ and W ′ are open subsets of half spaces? Here is the general definition: let f : V ′ → W ′, where V ′ ⊆ Rm and W ′ ⊆ Rn are arbitrary subsets. We say that f is smooth if for each p ∈ V ′ there exist an open neighborhood p ∈ Ṽ ′ ⊆ Rm and a smooth map f̃ : Ṽ ′ → Rn such that for each q ∈ V ′ ∩ Ṽ ′ we have that f (q) = f̃ (q). If M is a manifold with boundary, then its boundary ∂M is the subspace of points mapped to the boundary ∂H = {(0, p2, . . . , pn)} by some (and hence, it turns out, all) charts. The boundary is an (n − 1)dimensional manifold (without boundary). As an example, consider the closed ndimensional unit ball; it is an ndimensional manifold with boundary the unit (n − 1)sphere. Ordinary smooth manifolds correspond to manifolds with empty boundary. 2.4 Smooth Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Having defined smooth manifolds, we need to define smooth maps between them. No surprise: smoothness is a local question, so we may fetch the notion from Euclidean space by means of charts. (See Figure 2.9.) Definition 2.4.1 Let (M,A) and (N ,B) be smooth manifolds and p ∈ M . A continuous map f : M → N is smooth at p (or differentiable at p) if for any chart x : U → U ′ ∈ A with p ∈ U and any chart y : V → V ′ ∈ B with f (p) ∈ V the map y ◦ f U∩ f −1(V ) ◦ (x U∩ f −1(V ))−1 : x(U ∩ f −1(V ))→ V ′ is smooth at x(p). We say that f is a smooth map if it is smooth at all points of M . M x x (U ⊃ f –1(V )) y •p •f (p) f –1(V ) N VU f U′ V ′ Rm Rn •x (p) Figure 2.9. 2.4 Smooth Maps 33 Figure 2.9 will often find a less typographically challenging expression: “go up, over and down in the picture W f W−−−→ V x W ⏐⏐� y⏐⏐� x(W ) V ′, where W = U ∩ f −1(V ), and see whether you have a smooth map of open subsets of Euclidean spaces”. To see whether f in Definition 2.4.1 is smooth at p ∈ M you do not actually have to check all charts. Lemma 2.4.2 Let (M,A) and (N ,B) be smooth manifolds. A function f : M → N is smooth if (and only if) for all p ∈ M there exist charts (x,U ) ∈ A and (y, V ) ∈ B with p ∈ W = U ∩ f −1(V ) such that the composite y ◦ f W ◦ (x W )−1 : x(W )→ y(V ) is smooth. Exercise 2.4.3 Prove Lemma 2.4.2 (a full solution is provided in Appendix B, but you should really try yourself). Exercise 2.4.4 Show that the map R→ S1 sending p ∈ R to eip = (cos p, sin p) ∈ S1 is smooth. Exercise 2.4.5 Show that the map g : S2 → R4 given by g(p0, p1, p2) = (p1 p2, p0 p2, p0 p1, p20 + 2p21 + 3p22) defines a smooth injective map g̃ : RP2 → R4 via the formula g̃([p]) = g(p) (remember that p = 1; if you allow p ∈ R3−{0}, you should use g(p/p)). Exercise 2.4.6 Show that a map f : RPn → M is smooth if and only if the composite Sn g→ RPn f→ M is smooth, where g is the projection. Definition 2.4.7 A smooth map f : M → N is a diffeomorphism if it is a bijec tion, and the inverse is smooth too. Two smooth manifolds are diffeomorphic if there exists a diffeomorphism between them. 34 Smooth Manifolds Note 2.4.8 Note that this use of the word diffeomorphism coincides with the one used earlier for open subsets of Rn . Example 2.4.9 The smooth map R → R sending p ∈ R to p3 is a smooth homeomorphism, but it is not a diffeomorphism: the inverse is not smooth at 0 ∈ R. The problem is that the derivative is zero at 0 ∈ R: if a smooth bijective map f : R → R has a nowherevanishing derivative, then it is a diffeomorphism. The inverse function theorem, 4.2.1, gives the corresponding criterion for (local) smooth invertibility also in higher dimensions. Example 2.4.10 If a < b ∈ R, then the straight line f (t) = (b − a)t + a gives a diffeomorphism f : (0, 1)→ (a, b)with inverse given by f −1(t) = (t−a)/(b−a). Note that tan : (−π/2, π/2)→ R is a diffeomorphism. Hence all open intervals are diffeomorphic to the entire real line. Exercise 2.4.11 Show that RP1 and S1 are diffeomorphic. Exercise 2.4.12 Show that CP1 and S2 are diffeomorphic. Note 2.4.13 The distinction between differentiable and smooth of Definition 2.3.5 (i.e., whether there merely exists a smooth structure or one has been chosen) is not always relevant, but the reader may find pleasure in knowing that according to Kervaire and Milnor [10] the topological manifold S7 has 28 different smooth structures (up to “oriented” diffeomorphism, see Section 6.7 – 15 if orientation is ignored), and R4 has uncountably many [21]. As a side remark, one should notice that most physical situations involve differential equations of some sort, and so depend on the smooth struc ture, and not only on the underlying topological manifold. For instance, Baez remarks in This Week’s Finds in Mathematical Physics (Week 141), see www.classe.cornell.edu/spr/199912/msg0019934.html, that all of the 992 smooth structures on the 11sphere are relevant to string theory. Once one accepts the idea that there may be many smooth structures, one starts wondering what manifolds have a unique smooth structure (up to diffeomorphism). An amazing result in this direction recently appeared: Wang and Xu [22] have proved that the only odddimensional spheres with a unique smooth structure are S1, S3, S5 and S61(!). The evendimensional case is not fully resolved; S4 is totally mysterious, but apart from that one knows that S2, S6 and S56 are the only even dimensional spheres in a range of dimensions that support exactly one smooth 2.4 Smooth Maps 35 structure (at the time of writing it has been checked by Behrens, Hill, Hopkins and Ravenel, building on computations by Isaksen and Xu, up to dimension 140). Lemma 2.4.14 If f : (M,U) → (N ,V) and g : (N ,V) → (P,W) are smooth, then the composite g f : (M,U)→ (P,W) is smooth too. Proof. This is true for maps between Euclidean spaces, and we lift this fact to smooth manifolds. Let p ∈ M and choose appropriate charts x : U → U ′ ∈ U , such that p ∈ U , y : V → V ′ ∈ V , such that f (p) ∈ V , z : W → W ′ ∈W , such that g f (p) ∈ W . Then T = U ∩ f −1(V ∩ g−1(W )) is an open set containing p, and we have that zg f x−1x(T ) = (zgy−1)(y f x−1)x(T ), which is a composite of smooth maps of Euclidean spaces, and hence smooth. � In a picture, if S = V ∩ g−1(W ) and T = U ∩ f −1(S): T x T �� f T �� S yS �� gS �� W zW �� x(T ) y(S) z(W ). Going up and down with y does not matter. Exercise 2.4.15 Let f : M → X be a homeomorphism of topological spaces. If M is a smooth manifold then there is a unique smooth structure on X that makes f a diffeomorphism. In particular, note that, if M = X = R and f : M → X is the homeomor phism given by f (t) = t3, then the above gives a new smooth structure on R, but now (with respect to this structure) f : M → X is a diffeomorphism (as opposed to what was the case in Example 2.4.9), so the two smooth manifolds are diffeomorphic. Definition 2.4.16 Let (M,U) and (N ,V) be smooth manifolds. Then we let C∞(M, N ) = {smooth maps M → N } and C∞(M) = C∞(M,R). 36 Smooth Manifolds Note 2.4.17 A small digression, which may be disregarded if it contains words you haven’t heard before. The outcome of the discussion above is that we have a category C∞ of smooth manifolds: the objects are the smooth manifolds, and, if M and N are smooth, then C∞(M, N ) is the set of morphisms. The statement that C∞ is a category uses that the identity map is smooth (check), and that the composition of smooth functions is smooth, giving the composition in C∞: C∞(N , P)× C∞(M, N )→ C∞(M, P). The diffeomorphisms are the isomorphisms in this category. Definition 2.4.18 A smooth map f : M → N is a local diffeomorphism if for each p ∈ M there is an open set U ⊆ M containing p such that f (U ) is an open subset of N and f U : U → f (U ) is a diffeomorphism. Example 2.4.19 The projection Sn → RPn is a local diffeomorphism (Figure 2.10). Here is a more general example: let M be a smooth manifold, and i : M → M a diffeomorphism with the property that i(p) �= p, but i(i(p)) = p for all p ∈ M (such an animal is called a fixed point free involution). The quotient space M/ i gotten by identifying p and i(p) has a smooth structure, such that the projection Figure 2.10. Small open sets in RP2 correspond to unions U ∪ (–U), where U ⊆ S2 is an open set totally contained in one hemisphere. 2.5 Submanifolds 37 f : M → M/ i is a local diffeomorphism. We leave the proof of this claim as an exercise. Exercise 2.4.20 Show that M/ i has a smooth structure such that the projection f : M → M/ i is a local diffeomorphism. Exercise 2.4.21 If (M,U) is a smooth ndimensional manifold and p ∈ M , then there is a chart x : U → Rn such that x(p) = 0. Note 2.4.22 In differential topology one considers two smooth manifolds to be the same if they are diffeomorphic, and all properties one studies are unaffected by diffeomorphisms. Is it possible to give a classification of manifolds? That is, can we list all the smooth manifolds? On the face of it this is a totally overambitious question, but actually quite a lot is known, especially about the compact (Definition A.7.1) connected (Definition A.9.1) smooth manifolds. The circle is the only compact connected smooth 1manifold. In dimension two it is only slightly more interesting. As we discussed in Sec tion 1.5, you can obtain any compact (smooth) connected 2manifold by punching g holes in the sphere S2 and glue onto this either g handles or g Möbius bands. In dimension four and up total chaos reigns (and so it is here that most of the interesting stuff is to be found). Well, actually only the part within the parentheses is true in the last sentence: there is a lot of structure, much of it well understood. However, all of it is beyond the scope of this text. It involves quite a lot of manifold theory, but also algebraic topology and a subject called surgery, which in spirit is not so distant from the cutting and pasting techniques we used on surfaces in Section 1.5. For dimension three, the reader may refer back to Section 1.6.1. Note 2.4.23 The notion of a smooth map between manifolds with boundary is defined exactly as for ordinary manifolds, except that we need to use the extension of the notion of a smooth map V ′ → W ′ to cover the case where V ′ and W ′ are open subsets of half spaces as explained in Note 2.3.17. 2.5 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What should a smooth submanifold be? Well, for sure, Rn × {0} = {(t1, . . . , tn+k) ∈ Rn+k  tn+1 = · · · = tn+k = 0} ought to be a smooth submanifold of Rn × Rk = Rn+k , and – just as we modeled smooth manifolds locally by Euclidean space – we use this example to model the concept of a smooth submanifold (see Figure 2.11). At this point it is perhaps not entirely clear that this will cover all the examples we are interested in. However, we will see somewhat later (more precisely, in Theorem 4.7.4) that this definition is 38 Smooth Manifolds M N x x(U) = U′ U Rk Rn U′ ⊃ Rn Figure 2.11. equivalent to another, more conceptual and effectively checkable definition, which we as yet do not have all the machinery to formulate. Regardless, submanifolds are too important for us to afford to wait. Definition 2.5.1 Let (M,U) be a smooth (n + k)dimensional manifold. An ndimensional smooth submanifold in M (Figure 2.11) is a subset N ⊆ M such that for each p ∈ N there is a chart x : U → U ′ in U with p ∈ U such that x(U ∩ N ) = U ′ ∩ (Rn × {0}) ⊆ Rn × Rk . We say that the codimension of N in M is k. In this definition we identify Rn+k with Rn ×Rk . We often write Rn ⊆ Rn ×Rk instead of Rn × {0} ⊆ Rn × Rk to signify the subset of all points with the last k coordinates equal to zero. Note 2.5.2 The language of the definition really makes some sense: if (M,U) is a smooth manifold and N ⊆ M a submanifold, then N inherits a smooth structure such that the inclusion N → M is smooth. If p ∈ N choose a chart (x p,Up) on M with p ∈ Up such that x p(Up ∩ N ) = x p(Up)∩ (Rn × {0}). Restricting to Up ∩ N and projecting to the first n coordinates gives a homeomorphism from Up ∩ N to an open subset of Rn . On letting p vary we get a smooth atlas for N (the chart transformations consist of restrictions of chart transformations in U). Example 2.5.3 Let n be a natural number. Then Kn = {(p, pn)} ⊆ R2 is a smooth submanifold. We define a smooth chart x : R2 → R2, (p, q) → (p, q − pn). 2.5 Submanifolds 39 –2 –1 0 1 2 a0 –2 –1 0 1 2 a1 –2 –1 0 1 2 Figure 2.12. Note that as required, x is smooth, with smooth inverse given by (p, q) → (p, q + pn), and that x(Kn) = R1 × {0}. Exercise 2.5.4 Prove that S1 ⊂ R2 is a submanifold. More generally, prove that Sn ⊂ Rn+1 is a submanifold. Exercise 2.5.5 Show that the subset C ⊆ Rn+1 given by C = {(a0, . . . , an−1, t) ∈ Rn+1  tn + an−1tn−1 + · · · + a1t + a0 = 0}, a part of which is illustrated for n = 2 in Figure 2.12, is a smooth subman ifold. (Hint: express C as a graph of a realvalued smooth function and extend Example 2.5.3 to cover such graphs in general.) Exercise 2.5.6 The subset K = {(p, p)  p ∈ R} ⊆ R2 is not a smooth submanifold. Note 2.5.7 If N ⊆ M is a smooth submanifold and dim(M) = dim(N ) then N ⊆ M is an open subset (called an open submanifold). Otherwise dim(M) > dim(N ). Example 2.5.8 Let MnR be the set of n × n matrices. This is a smooth manifold since it is homeomorphic to Rn 2 . The subset GLn(R) ⊆ MnR of invertible matrices is an open submanifold (the determinant function det : MnR→ R is continuous, so the inverse image GLn(R) = det−1(R− {0}) of the open set R \ {0} ⊆ R is open). 40 Smooth Manifolds Exercise 2.5.9 If V is an ndimensional vector space, let GL(V ) be the set of linear isomorphisms α : V ∼= V . By representing any linear isomorphism of Rn in terms of the standard basis, we may identify GL(Rn) and GLn(R). Any linear isomorphism f : V ∼= W gives a bijection GL( f ) : GL(V ) ∼= GL(W ) sending α : V ∼= V to f α f −1 : W ∼= W . Hence, any linear isomorphism f : V ∼= Rn (i.e., a choice of basis) gives a bijection GL( f ) : GL(V ) ∼= GLnR, and hence a smooth manifold structure on GL(V ) (with a diffeomorphism to the open subset GLnR of Euclidean n2space). Prove that the smooth structure on GL(V ) does not depend on the choice of f : V ∼= Rn . If h : V ∼= W is a linear isomorphism, prove that GL(h) : GL(V ) ∼= GL(W ) is a diffeomorphism respecting composition and the identity element. Example 2.5.10 Let Mm×nR be the set of m × n matrices (if m = n we write Mn(R) instead of Mn×n(R)). This is a smooth manifold since it is homeomorphic to Rmn . Let 0 ≤ r ≤ min(m, n). That a matrix has rank r means that it has an r × r invertible submatrix, but no larger invertible submatrices. The subset Mrm×n(R) ⊆ Mm×nR of matrices of rank r is a submanifold of codi mension (n− r)(m− r). Since some of the ideas will be valuable later on, we spell out a proof. For the sake of simplicity, we treat the case where our matrices have an invertible r × r submatrix in the upper lefthand corner. The other cases are covered in a similar manner, taking care of indices (or by composing the chart we give below with a diffeomorphism on Mm×nR given by multiplying with permutation matrices so that the invertible submatrix is moved to the upper lefthand corner). So, consider the open set U of matrices X = [ A B C D ] with A ∈ Mr (R), B ∈ Mr×(n−r)(R), C ∈ M(m−r)×r (R) and D ∈ M(m−r)×(n−r)(R) such that det(A) �= 0 (i.e., such that A is in the open subset GLr (R) ⊆ Mr (R)). The matrix X has rank exactly r if and only if the last n−r columns are in the span of the first r . Writing this out, this means that X is of rank r if and only if there is an r × (n − r) matrix T such that[ B D ] = [ A C ] T, which is equivalent to T = A−1 B and D = C A−1 B. Hence U ∩ Mrm×n(R) = {[ A B C D ] ∈ U ∣∣∣∣ D − C A−1 B = 0} . The map U →GLr (R)× Mr×(n−r)(R)× M(m−r)×r (R)× M(m−r)×(n−r)(R), 2.5 Submanifolds 41[ A B C D ] → (A, B,C, D − C A−1 B) is a diffeomorphism onto an open subset of Mr (R)×Mr×(n−r)(R)×M(m−r)×r (R)× M(m−r)×(n−r)(R) ∼= Rmn , and therefore gives a chart having the desired property that U ∩Mrm×n(R) is the set of points such that the last (m− r)(n− r) coordinates vanish. Definition 2.5.11 A smooth map f : N → M is an imbedding if the image f (N ) ⊆ M is a submanifold, and the induced map N → f (N ) is a diffeomorphism. Exercise 2.5.12 Prove that C→ M2(R), x + iy → [ x −y y x ] defines an imbedding. More generally it defines an imbedding Mn(C)→ Mn(M2(R)) ∼= M2n(R). Show also that this imbedding sends “conjugate transpose” to “transpose” and “multiplication” to “multiplication”. Exercise 2.5.13 Show that the map f : RPn → RPn+1, [p] = [p0, . . . , pn] → [p, 0] = [p0, . . . , pn, 0] is an imbedding. Note 2.5.14 Later we will give a very efficient way of creating smooth subman ifolds, getting rid of all the troubles of finding actual charts that make the subset look like Rn in Rn+k . We shall see that if f : M → N is a smooth map and q ∈ N then more often than not the inverse image f −1(q) = {p ∈ M  f (p) = q} is a submanifold of M . Examples of such submanifolds are the sphere and the space of orthogonal matrices (the inverse image of the identity matrix under the map sending a matrix A to AT A, where AT is A transposed). Example 2.5.15 This is an example where we have the opportunity to use a bit of topology. Let f : M → N be an imbedding, where M is a (nonempty) compact ndimensional smooth manifold and N is a connected ndimensional smooth man ifold. Then f is a diffeomorphism. This is so because f (M) is compact, and hence 42 Smooth Manifolds closed, and open since it is a codimensionzero submanifold. Hence f (M) = N since N is connected. But since f is an imbedding, the map M → f (M) = N is – by definition – a diffeomorphism. Exercise 2.5.16 (This is an important exercise. Do it: you will need the result several times.) Let i1 : N1 → M1 and i2 : N2 → M2 be smooth imbeddings and let f : N1 → N2 and g : M1 → M2 be continuous maps such that i2 f = gi1 (i.e., the diagram N1 f−−−→ N2 i1 ⏐⏐� i2⏐⏐� M1 g−−−→ M2 commutes). Show that if g is smooth, then f is smooth. Exercise 2.5.17 Show that the composite of two imbeddings is an imbedding. Exercise 2.5.18 Let 0 < m ≤ n and define the Milnor manifold by H(m, n) = { ([p], [q]) ∈ RPm × RPn  m∑ k=0 pkqk = 0 } . Prove that H(m, n) ⊆ RPm × RPn is a smooth (m + n − 1)dimensional submanifold. Note 2.5.19 The Milnor manifolds and their complex counterparts are particu larly important manifolds, because they in a certain sense give the building blocks for all manifolds (up to a certain equivalence relation called cobordism). 2.6 Products and Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition 2.6.1 Let (M,U) and (N ,V) be smooth manifolds. The (smooth) product is the smooth manifold you get by giving the product M × N the smooth atlas given by the charts x × y : U × V →U ′ × V ′, (p, q) → (x(p), y(q)), where (x,U ) ∈ U and (y, V ) ∈ V . Exercise 2.6.2 Check that this definition makes sense. 2.6 Products and Sums 43 Figure 2.13. The torus is a product. The bolder curves in the illustration try to indicate the submanifolds {1} × S1 and S1 × {1}. Note 2.6.3 Even if the atlases we start with are maximal, the charts of the form x × y do not form a maximal atlas on the product, but as always we can consider the associated maximal atlas. Example 2.6.4 We know a product manifold already: the torus S1 × S1 (Figure 2.13). Exercise 2.6.5 Show that the projection pr1 : M × N → M, (p, q) → p is a smooth map. Choose a point p ∈ M . Show that the map i p : N → M × N , q → (p, q) is an imbedding. Exercise 2.6.6 Show that giving a smooth map Z → M×N is the same as giving a pair of smooth maps Z → M and Z → N . Hence we have a bijection C∞(Z ,M × N ) ∼= C∞(Z ,M)× C∞(Z , N ). Exercise 2.6.7 Show that the infinite cylinder R1×S1 is diffeomorphic to R2\{0}. See Figure 2.14. More generally: R1 × Sn is diffeomorphic to Rn+1 \ {0}. Exercise 2.6.8 Let f : M → M ′ and g : N → N ′ be imbeddings. Then f × g : M × N → M ′ × N ′ is an imbedding. 44 Smooth Manifolds Figure 2.14. Looking down into the infinite cylinder. Exercise 2.6.9 Show that there exists an imbedding Sn1 × · · · × Snk → R1+∑ki=1 ni . Exercise 2.6.10 Why is the multiplication of matrices GLn(R)× GLn(R)→ GLn(R), (A, B) → A · B a smooth map? This, together with the existence of inverses, makes GLn(R) a “Lie group”. For the record: a Lie group is a smooth manifold M with a smooth “multiplica tion” M ×M → M that is associative, and it has a neutral element and all inverses (in GLn(R) the neutral element is the identity matrix). Exercise 2.6.11 Why is the multiplication S1 × S1 → S1, (eiθ , eiτ ) → eiθ · eiτ = ei(θ+τ) a smooth map? This is our second example of a Lie group. Definition 2.6.12 Let (M,U) and (N ,V) be smooth manifolds. The (smooth) disjoint union (or sum) is the smooth manifold you get by giving the disjoint union M ∐ N the smooth structure given by U ∪ V . See Figure 2.15. Exercise 2.6.13 Check that this definition makes sense. Note 2.6.14 As for the product, the atlas we give the sum is not maximal (a chart may have disconnected source and target). There is nothing wrong a priori with taking the disjoint union of an mdimensional manifold with an ndimensional manifold. The result will of course be neither m nor ndimensional. Such examples will not be important to us, and you will find that in arguments we may talk about a smooth manifold, and without hesitation later on start talking about its dimension. This is justified since we can consider one component at a time, and each component will have a welldefined dimension. 2.6 Products and Sums 45 Figure 2.15. The disjoint union of two tori (imbedded in R3). Figure 2.16. Note 2.6.15 Manifolds with boundary, as defined in Note 2.3.17, do not behave nicely under products, as can be seen already from the simplest example [0, 1] × [0, 1]. There are ways of “rounding the corners” that have been used in the literature to deal with this problem, but it soon becomes rather technical. There is no similar problem with the disjoint union. Example 2.6.16 The Borromean rings (Figure 2.16) give an interesting example showing that the imbedding in Euclidean space is irrelevant to the manifold: the Borromean rings amount to the disjoint union of three circles S1 ∐ S1 ∐ S1. Don’t get confused: it is the imbedding in R3 that makes your mind spin: the manifold itself is just three copies of the circle! Moral: an imbedded manifold is something more than just a manifold that can be imbedded. Exercise 2.6.17 Prove that the inclusion inc1 : M ⊂ M ∐ N is an imbedding. 46 Smooth Manifolds Exercise 2.6.18 Show that giving a smooth map M ∐ N → Z is the same as giving a pair of smooth maps M → Z and N → Z . Hence we have a bijection C∞(M ∐ N , Z) ∼= C∞(M, Z)× C∞(N , Z). Exercise 2.6.19 Find all the errors in the hints in Appendix B for the exercises from Chapter 2. 3 The Tangent Space In this chapter we will study linearizations. You have seen this many times before as tangent lines and tangent planes (for curves and surfaces in Euclidean space), and the main difficulty you will encounter is that the linearizations must be defined intrinsically – i.e., in terms of the manifold at hand – and not with reference to some big ambient space. We will shortly (in Predefinition 3.0.5) give a simple and perfectly fine technical definition of the tangent space, but for future convenience we will use the concept of germs in our final definition. This concept makes nota tion and bookkeeping easy and is good for all things local (in the end it will turn out that due to the existence of socalled smooth bump functions (see Section 3.2) we could have stayed global in our definitions). An important feature of the tangent space is that it is a vector space, and a smooth map of manifolds gives a linear map of vector spaces. Eventually, the chain rule expresses the fact that the tangent space is a “natural” construction (which actually is a very precise statement that will reappear several times in different contexts. It is the hope of the author that the reader, through the many examples, in the end will appreciate the importance of being natural – as well as earnest). Beside the tangent space, we will also briefly discuss its sibling, the cotangent space, which is concerned with linearizing the space of realvalued functions, and which is the relevant linearization for many applications. Another interpretation of the tangent space is as the space of derivations, and we will discuss these briefly since they figure prominently in many expositions. They are more abstract and less geometric than the path we have chosen – as a matter of fact, in our presentation derivations are viewed as a “double dualization” of the tangent space. 3.0.1 The Idea of the Tangent Space of a Submanifold of Euclidean Space Given a submanifold M of Euclidean space Rn , it is fairly obvious what we should mean by the “tangent space” of M at a point p ∈ M . In purely physical terms, the tangent space should be the following subspace of Rn . If a particle moves on some curve in M and at p suddenly “loses its grip on M” (Figure 3.1) it will continue out in the ambient space along a straight line (its “tangent”). This straight line is determined by its velocity vector at the point 48 The Tangent Space Figure 3.1. A particle loses its grip on M and flies out on a tangent Figure 3.2. A part of the space of all tangents where it flies out into space. The tangent space should be the linear subspace of Rn containing all these vectors. See Figure 3.2. When talking about manifolds it is important to remember that there is no ambient space to fly out into, but we still may talk about a tangent space. 3.0.2 Partial Derivatives The tangent space is all about the linearization in Euclidean space. To fix notation we repeat some multivariable calculus. Definition 3.0.1 Let f : U → R be a function where U is an open subset of Rn containing p = (p1, . . . pn). The i th partial derivative of f at p is the number (if it exists) Di f (p) = Di p f = lim h→0 1 h ( f (p + hei )− f (p)), where ei is the i th unit vector ei = (0, . . . , 0, 1, 0, . . . , 0) (with a 1 in the i th coordinate). We collect the partial derivatives in a 1× n matrix The Tangent Space 49 D f (p) = Dp f = (D1 f (p), . . . , Dn f (p)). Definition 3.0.2 If f = ( f1, . . . , fm) : U → Rm is a function where U is an open subset of Rn containing p = (p1, . . . pn), then the Jacobian matrix is the m × n matrix D f (p) = Dp( f ) = ⎡⎢⎣D f1(p)... D fm(p) ⎤⎥⎦ . In particular, if g = (g1, . . . gm) : (a, b)→ Rm the Jacobian is an m × 1 matrix, or element in Rm , which we write as g′(c) = Dg(c) = ⎡⎢⎣g ′ 1(c) ... g′m(c) ⎤⎥⎦ ∈ Rm . For convenience, we cite the “flat” (i.e., in Euclidean space) chain rule. For a proof, see, e.g., Section 29 of [19], or any decent book on multivariable calculus. Lemma 3.0.3 (The Flat Chain Rule) Let g : (a, b) → U and f : U → R be smooth functions where U is an open subset of Rn and c ∈ (a, b). Then ( f g)′(c) = D( f )(g(c)) · g′(c) = n∑ j=1 D j f (g(c)) · g′j (c). Note 3.0.4 When considered as a vector space, we insist that the elements in Rn are standing vectors (so that linear maps can be represented by multiplication by matrices from the left); when considered as a manifold the distinction between lying and standing vectors is not important, and we use either convention as may be typographically convenient. It is a standard fact from multivariable calculus (see, e.g., Section 28 of [19]) that if f : U → Rm is continuously differentiable at p (all the partial derivatives exist and are continuous at p), where U is an open subset of Rn , then the Jacobian is the matrix associated (in the standard bases) with the unique linear transformation L : Rn → Rm such that lim h→0 1 h( f (p + h)− f (p)− L(h)) = 0. Predefinition 3.0.5 (of the Tangent Space) Let M be a smooth manifold, and let p ∈ M. Consider the set of all curves γ : R → M with γ (0) = p. On this set we define the following equivalence relation: given two curves γ : R → M and 50 The Tangent Space γ1 : R→ M with γ (0) = γ1(0) = p, we say that γ and γ1 are equivalent if for all charts x : U → U ′ with p ∈ U we have an equality of vectors (xγ )′(0) = (xγ1)′(0). Then the tangent space of M at p is the set of all equivalence classes. There is nothing wrong with this definition, in the sense that it is naturally iso morphic to the one we are going to give in a short while (see Definition 3.3.1). However, in order to work efficiently with our tangent space, it is fruitful to intro duce some language. It is really not necessary for our curves to be defined on all of R, but on the other hand it is not important to know the domain of definition as long as it contains a neighborhood around the origin. 3.1 Germs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Whatever one’s point of view on tangent vectors is, it is a local concept. The tangent of a curve passing through a given point p is only dependent upon the behavior of the curve close to the point. Hence it makes sense to divide out by the equivalence rela