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Introduction to Differentiable Manifolds INTRODUCTION TO DIFFERENTIABLE MANIFOLDS spring 2012 Course by Prof. dr. R.C.A.M. van der Vorst Solution manual Dr. G.J. Ridderbos 1 2 Introduction to differentiable manifolds Lecture notes version 2.1, November 5, 2012 This is a self contained set of lecture notes. The notes were written by Rob van der Vorst. The solution manual is written by Guit-Jan Ridderbos. We follow the book ‘Introduction to Smooth Manifolds’ by John M. Lee as a reference text [1]. Additional reading and exercises are take from ‘An introduction to manifolds’ by Loring W. Tu [2]. This document was produced in LATEX and the pdf-file of these notes is available on the following website www.few.vu.nl/˜vdvorst They are meant to be freely available for non-commercial use, in the sense that “free software” is free. More precisely: This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/2.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA. CONTENTS References 5 I. Manifolds 6 1. Topological manifolds 6 2. Differentiable manifolds and differentiable structures 13 3. Immersions, submersions and embeddings 20 II. Tangent and cotangent spaces 32 4. Tangent spaces 32 5. Cotangent spaces 38 6. Vector bundles 41 6.1. The tangent bundle and vector fields 44 6.2. The cotangent bundle and differential 1-forms 46 III. Tensors and differential forms 50 7. Tensors and tensor products 50 8. Symmetric and alternating tensors 55 3 9. Tensor bundles and tensor fields 61 10. Differential forms 66 11. Orientations 69 IV. Integration on manifolds 75 12. Integrating m-forms on Rm 75 13. Partitions of unity 76 14. Integration on of m-forms on m-dimensional manifolds. 81 15. The exterior derivative 87 16. Stokes’ Theorem 92 V. De Rham cohomology 97 17. Definition of De Rham cohomology 97 18. Homotopy invariance of cohomology 98 VI. Exercises 102 Manifolds Topological Manifolds 102 Differentiable manifolds 103 Immersion, submersion and embeddings 104 Tangent and cotangent spaces Tangent spaces 106 Cotangent spaces 107 Vector bundles 108 Tensors Tensors and tensor products 109 Symmetric and alternating tensors 109 Tensor bundles and tensor fields 110 Differential forms 111 Orientations 111 Integration on manifolds Integrating m-forms on Rm 112 Partitions of unity 112 Integration of m-forms on m-dimensional manifolds 112 The exterior derivative 113 Stokes’ Theorem 114 4 Extra’s 115 De Rham cohomology Definition of De Rham cohomology 117 VII. Solutions 118 5 References [1] John M. Lee. Introduction to smooth manifolds, volume 218 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2003. [2] Loring W. Tu. An introduction to manifolds. Universitext. Springer, New York, second edition, 2011. 6 I. Manifolds 1. Topological manifolds Basically an m-dimensional (topological) manifold is a topological space M which is locally homeomorphic to Rm. A more precise definition is: Definition 1.1. 1 A topological space M is called an m-dimensional (topological) manifold, if the following conditions hold: (i) M is a Hausdorff space, (ii) for any p M there exists a neighborhood2 U of p which is homeomorphic 2 to an open subset V Rm, and ⇢ (iii) M has a countable basis of open sets. Axiom (ii) is equivalent to saying that p M has a open neighborhood U p 2 3 homeomorphic to the open disc Dm in Rm. We say M is locally homeomorphic to Rm. Axiom (iii) says that M can be covered by countably many of such neighbor- hoods. FIGURE 1. Coordinate charts (U,j). 1See Lee, pg. 3. 2Usually an open neighborhood U of a point p M is an open set containing p. A neighborhood 2 of p is a set containing an open neighborhood containing p. Here we will always assume that a neighborhood is an open set. 7 FIGURE 2. The transition maps jij. Recall some notions from topology: A topological space M is called Hausdorff if for any pair p,q M, there exist (open) neighborhoods U p, and U q such 2 3 0 3 that U U0 = ?. For a topological space M with topology t, a collection b t is \ ⇢ a basis if and only if each U t can be written as union of sets in b. A basis is 2 called a countable basis if b is a countable set. Figure 1 displays coordinate charts (U,j), where U are coordinate neighbor- hoods, or charts, and j are (coordinate) homeomorphisms. Transitions between 1 different choices of coordinates are called transitions maps j = j j− , which ij j ◦ i are again homeomorphisms by definition. We usually write x = j(p), j : U n 1 1 ! V R , as coordinates for U, see Figure 2, and p = j− (x), j− : V U M, as ⇢ ! ⇢ a parametrization of U. A collection A = (ji,Ui) i I of coordinate charts with { } 2 M = U , is called an atlas for M. [i i The following Theorem gives a number of useful characteristics of topological manifolds. Theorem 1.4. 3 A manifold is locally connected, locally compact, and the union of countably many compact subsets. Moreover, a manifold is normal and metrizable. J 1.5 Example. M = Rm; the axioms (i) and (iii) are obviously satisfied. As for (ii) we take U = Rm, and j the identity map. I 3Lee, 1.6, 1.8, and Boothby. 8 1 2 2 2 1.6 Example. M = S = p =(p1, p2) R p + p = 1 ; as before (i) and (iii) J { 2 | 1 2 } are satisfied. As for (ii) we can choose many different atlases. (a): Consider the sets U = p S1 p > 0 , U = p S1 p < 0 , 1 { 2 | 2 } 2 { 2 | 2 } U = p S1 p < 0 , U = p S1 p > 0 . 3 { 2 | 1 } 4 { 2 | 1 } The associated coordinate maps are j1(p)=p1, j2(p)=p1, j3(p)=p2, and j4(p)=p2. For instance 1 2 j− (x)= x, 1 x , 1 − ⇣ p ⌘ 1 and the domain is V =( 1,1). It is clear that j and j− are continuous, and 1 − i i therefore the maps ji are homeomorphisms. With these choices we have found an atlas for S1 consisting of four charts. (b): (Stereographic projection) Consider the two charts U = S1 (0,1) , and U = S1 (0, 1) . 1 \{ } 2 \{ − } The coordinate mappings are given by 2p 2p j (p)= 1 , and j (p)= 1 , 1 1 p 2 1 + p − 2 2 which are continuous maps from Ui to R. For example 2 1 4x x 4 j− (x)= , − , 1 x2 + 4 x2 + 4 ⇣ ⌘ which is continuous from R to U1. FIGURE 3. The stereographic projection describing j1. (c): (Polar coordinates) Consider the two charts U = S1 (1,0) , and U = 1 \{ } 2 S1 ( 1,0) . The homeomorphism are j (p)=q (0,2p) (polar angle counter \{ − } 1 2 9 clockwise rotation), and j2(p)=q ( p,p) (complex argument). For example 1 2 − j1− (q)=(cos(q),sin(q)). I n n+1 2 1.8 Example. M = S = p =(p1, pn+1) R p = 1 . Obvious exten- J { ··· 2 || | } sion of the choices for S1. I 1.9 Example. (see Lee) Let U Rn be an open set and g : U Rm a continuous J ⇢ ! function. Define n m M = G(g)= (x,y) R R : x U, y = g(x) , { 2 ⇥ 2 } endowed with the subspace topology, see 1.17. This topological space is an n- dimensional manifold. Indeed, define p : Rn Rm Rn as p(x,y)=x, and ⇥ ! j = p , which is continuous onto U. The inverse j 1(x)=(x,g(x)) is also |G(g) − continuous. Therefore (G(g),j) is an appropriate atlas. The manifold G(g) is { } homeomorphic to U. I J 1.10 Example. M = PRn, the real projective spaces. Consider the following equivalence relation on points on Rn+1 0 : For any two x,y Rn+1 0 define \{ } 2 \{ } x y if there exists a l = 0, such that x = ly. ⇠ 6 Define PRn = [x] : x Rn+1 0 as the set of equivalence classes. One can { 2 \{ }} think of PRn as the set of lines through the origin in Rn+1. Consider the natural n+1 n n 1 map p : R 0 PR , via p(x)=[x]. A set U PR is open if p− (U) is open \{ }! ⇢ in Rn+1 0 . This makes p a quotient map and p is continuous. The equivalence \{ } relation is open (see [2, Sect. 7.6]) and therefore PRn Hausdorff and second ⇠ countable, see 1.17. Compactness of PRn can be proves by showing that there exists a homeomorphism from PRn to Sn/ , see [2, Sect. 7.6]. In order to verify ⇠ that we are dealing with an n-dimensional manifold we need to describe an atlas n n+1 for PR . For i = 1, n + 1, define Vi R 0 as the set of points x for which ··· ⇢ \{ } x = 0, and define U = p(V ). Furthermore, for any [x] U define i 6 i i 2 i x1 xi 1 xi+1 xn+1 ji([x]) = , , − , , , , xi ··· xi xi ··· xi ⇣ ⌘ which is continuous. This follows from the fact that j p is continuous. The ◦ continuous inverse is given by 1 ji− (z1, ,zn)= (z1, ,zi 1,1,zi, ,zn) . ··· ··· − ··· n ⇥ ⇤ 1 These charts Ui cover PR . In dimension n = 1 we have that PR ⇠= S , and in the dimension n = 2 we obtain an immersed surface PR2 as shown in Figure 4.
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