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Manifolds Bruno Martelli Manifolds Bruno Martelli Dipartimento di Matematica, Largo Pontecorvo 5, 56127 Pisa, Italy E-mail address: martelli at dm dot unipi dot it version: May 30, 2017 Contents Introduction 1 Chapter 1. Preliminaries 3 1.1. General topology 3 1.2. Algebraic topology 8 1.3. Multivariable analysis 12 1.4. Projective geometry 16 Chapter 2. Tensors 19 2.1. Multilinear algebra 19 2.2. Tensors 25 2.3. Scalar products 33 2.4. The symmetric and exterior algebras 35 2.5. Grassmannians 42 2.6. Orientation 43 Chapter 3. Smooth manifolds 45 3.1. Smooth manifolds 45 3.2. Smooth maps 49 3.3. Partitions of unity 50 3.4. Tangent space 54 3.5. Smooth coverings 59 3.6. Orientation 62 3.7. Submanifolds 67 3.8. Immersions, embeddings, and submersions 68 3.9. Examples 73 3.10. Homotopy and isotopy 77 3.11. The Whitney embedding 79 Chapter 4. Bundles 85 4.1. Fibre bundles 85 4.2. Vector bundles 87 4.3. Tangent bundle 90 4.4. Sections 93 4.5. Riemannian metric 97 v vi CONTENTS Chapter 5. The basic toolkit 103 5.1. Vector fields 103 5.2. Flows 105 5.3. Ambient isotopy 107 5.4. Lie brackets 109 5.5. Foliations 115 5.6. Tubular neighbourhoods 118 5.7. Transversality 123 5.8. Manifolds with boundary 126 5.9. Cut and paste 132 Chapter 6. Differential forms 137 6.1. Differential forms 137 6.2. Integration 141 6.3. Stokes’ Theorem 146 Chapter 7. De Rham cohomology 153 7.1. Definition 153 7.2. The Poincaré Lemma 157 7.3. The Mayer – Vietoris sequence 161 7.4. Compactly supported forms 167 7.5. Poincaré duality 173 7.6. Intersection theory 180 Chapter 8. Riemannian manifolds 185 8.1. The metric tensor 185 8.2. Connections 188 8.3. The Levi–Civita connection 194 8.4. Geodesics 199 8.5. Completeness 207 8.6. Curvature 209 Introduction Copyright notices. The text is released under the Creative Commons- BY-NC-SA license. You are allowed to distribute, modify, adapt, and use of this work for non-commercial purposes, as long as you correctly attribute its authorship and keep using the same license. The pictures used here are all in the public domain (both those created by the author and those downloaded from Wikipedia, which were already in the public domain). except the following ones that are released with a CC-BY-SA license and can be downloaded from Wikipedia: • Figure 1.2 (the circle is a manifold), created by KSmrq. • Figures 5.1 (vector fields on a torus) and 8.2 (the exponential map), created by RokerHRO. • Figure 5.11 (the connected sum) created by Oleg Alexandrov, The painting in the front page is a courtesy of Mariette Michelle Egreteau. 1 CHAPTER 1 Preliminaries We state here some basic notions of topology and analysis that we will use in this book. The proofs of some theorems are omitted and can be found in many excellent sources. 1.1. General topology 1.1.1. Topological spaces. A topological space is a pair (X; τ) where X is a set and τ is a collection of subsets of X called open subsets, satisfying the following axioms: • ? and X are open subsets; • the arbitrary union of open subsets is an open subset; • the finite intersection of open subsets is an open subset. The complement X nU of an open subset U 2 τ is called closed. When we denote a topological space, we often write X instead of (X; τ) for simplicity. A neighbourhood of a point x 2 X is any subset N ⊂ X containing an open set U that contains x, that is x 2 U ⊂ N ⊂ X. 1.1.2. Examples. There are many ways to construct topological spaces and we summarise them here very briefly. Metric spaces. Every metric space (X; d) is also naturally a topological space: by definition, a subset U ⊂ X is open () for every x0 2 U there is an r > 0 such that the open ball B(x0; r) = x 2 X d(x; x0) < r is entirely contained in U. n In particular R is a topological space, whose topology is induced by the euclidean distance between points. Q Product topology. The cartesian product X = i2I Xi of two or more topological spaces is a topological space: by definition, a subset U ⊂ X is Q open () it is a (possibly infinite) union of products i2I Ui of open subsets Ui ⊂ Xi , where Ui 6= Xi only for finitely many i. This is the coarsest topology (that is, the topology with the fewest open sets) on X such that the projections X ! Xi are all continuous. 3 4 1. PRELIMINARIES Subspace topology. Every subset S ⊂ X of a topological space X is also naturally a topological space: by definition a subset U ⊂ S is open () there is an open subset V ⊂ X such that U = V \ S. This is the coarsest topology on S such that the inclusion i : S,! X is continuous. n In particular every subset S ⊂ R is naturally a topological space. Quotient topology. Let f : X ! Y be a surjective map. A topology on X induces one on Y as follows: by definition a set U ⊂ Y is open () its counterimage f −1(U) is open in X. This is the finest topology (that is, the one with the most open subsets) on Y such that the map f : X ! Y is continuous. A typical situation is when Y is the quotient space Y = X=∼ for some equivalence relation ∼ on X, and X ! Y is the induced projection. 1.1.3. Continuous maps. A map f : X ! Y between topological spaces is continuous if the inverse of every open subset of Y is an open subset of X. The map f is a homeomorphism if it has an inverse f −1 : Y ! X which is also continuous. Two topological spaces X and Y are homeomorphic if there is a homeomor- phism f : X ! Y relating them. Being homeomorphic is clearly an equivalence relation. 1.1.4. Reasonable assumptions. A topological space can be very wild, but most of the spaces encountered in this book will satisfy some reasonable assumptions, that we now list. Hausdorff. A topological space X is Hausdorff if every two distinct points x; y 2 X have disjoint open neighbourhoods Ux and Uy , that is Ux \ Uy = ?. n The euclidean space R is Hausdorff. Products and subspaces of Hausdorff spaces are also Hausdorff. Second-countable. A base for a topological space X is a set of open sub- sets fUi g such that every open set is an arbitrary union of these. A topological space X is second-countable if it has a countable base. n The euclidean space R is second-countable. Countable products and subspaces of second-countable spaces are also second-countable. Connected. A topological space X is connected if it is not the disjoint union X = X1 t X2 of two non-empty open subsets X1;X2. Every topologi- cal space X is partitioned canonically into maximal connected subsets, called connected components. Given this canonical decomposition, it is typically harmless to restrict our attention to connected spaces. A slightly stronger notion is that of path-connectedness. A space X is path-connected if for every x; y 2 X there is a path connecting them, that is 1.1. GENERAL TOPOLOGY 5 a continuous map α: [0; 1] ! X with α(0) = x and α(1) = y. Every path- connected space is connected. The converse is also true if one assumes the reasonable assumption that the topological space we are considering is locally path-connected, that is every point has a path-connected neighbourhood. n The Euclidean space R is path-connected. Products and quotients of (path-)connected spaces are (path-)connected. Locally compact. A topological space X is locally compact if every point n x 2 X has a compact neighbourhood. The euclidean space R is locally compact. 1.1.5. Reasonable consequences. The reasonable assumptions listed in the previous section have some nice and reasonable consequences. Countable base with compact closure. We first note the following. Proposition 1.1.1. If a topological space X is Hausdorff and locally com- pact, every x 2 X has an open neighbourhood U(x) with compact closure. Proof. Every x 2 X has a compact neighbourhood V (x), that is closed since X is Hausdorff. The neighbourhood V (x) contains an open neighbour- hood U(x) of x, whose closure is contained in V (x) and hence compact. Proposition 1.1.2. Every locally compact second-countable Hausdorff space X has a countable base made of open sets with compact closure. Proof. Let fUi g be a countable base. For every open set U ⊂ X and x 2 U, there is an open neighbourhood U(x) ⊂ U of x with compact closure, which contains a Ui that contains x. Therefore the Ui with compact closure suffice as a base for X. Exhaustion by compact sets. Let X be a topological space. An exhaus- tion by compact subsets is a countable family K1;K2;::: of compact subsets such that Ki ⊂ int(Ki+1) for all i and [i Ki = X. n The standard example is the exhaustion of R by closed balls n Ki = B(0; i) = x 2 R kxk ≤ i : Proposition 1.1.3. Every locally compact second-countable Hausdorff space X has an exhaustion by compact subsets. Proof. The space X has a countable base U1;U2;::: of open sets with compact closures.
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