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Differential Geometry

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Differential Geometry DIFFERENTIAL GEOMETRY RUI LOJA FERNANDES Date: May 11, 2021. 1 Contents Part 1. Basic Concepts 5 0. Manifolds as subsets of Euclidean space 7 1. Abstract Manifolds 12 2. Manifolds with Boundary 19 3. Partitions of Unity 23 4. The Tangent Space 27 5. The Differential 35 6. Immersions, Submersions and Submanifolds 38 7. Embeddings and Whitney’s Theorem 47 8. Foliations 55 9. Quotients 64 Part 2. Lie Theory 74 10. Vector Fields and Flows 75 11. Lie Bracket and Lie Derivative 83 12. Distributions and the Frobenius Theorem 88 13. Lie Groups and Lie Algebras 93 14. Integrations of Lie Algebras 100 15. The Exponential Map 106 16. Groups of Transformations 110 Part 3. Differential Forms 120 17. Differential Forms and Tensor Fields 122 18. Differential and Cartan Calculus 132 19. Integration on Manifolds 138 20. de Rham Cohomology 144 21. The de Rham Theorem 150 22. Homotopy Invariance and Mayer-Vietoris Sequence 158 23. Computations in Cohomology 168 24. The Degree and the Index 178 Part 4. Fiber Bundles 186 25. Vector Bundles 188 26. The Thom Class and the Euler Class 197 27. Pullbacks of Vector Bundles 207 28. The Classification of Vector Bundles 213 29. Connections and Parallel Transport 217 30. Curvature and Holonomy 225 31. The Chern-Weil homomorphism 231 32. Characteristic Classes 238 33. Fiber Bundles 248 34. Principal Fiber Bundles 255 2 Preface These are lecture notes for the courses “Differentiable Manifolds I” and “Differentiable Manifolds II”, that I am lecturing at UIUC. This course is usually taken by graduate students in Mathematics in their first or second year of studies. The background for this course is a basic knowledge of analysis, algebra and topology. My main aim in writing up these lectures notes is to offer a written version of the lectures. This should give a chance to students to concentrate more on the class, without worrying about taking notes. It offers also a guide for what material was covered in class. These notes do not replace the recommended texts for this course, quite the contrary: I hope they will be a stimulus for the students to consult those works. In fact, some of these notes follow the material in theses texts. These notes are organized into sections, each of these should correspond to lectures of approximately to 1 hour and 30 minutes of classroom time. However, some sections do include more material than others, which corre- spond to different rhythms in class. The exercises at the end of each section are a very important part of the course, since one learns a good deal about mathematics by solving exercises. Moreover, sometimes the exercises con- tain results that were mentioned in class, but not proved, and which are used in later sections. The students should also keep in mind that the exercises are not homogeneous: this is in line with the fact that in mathematics when one faces for the first time a problem, one usually does not know if it has an easy solution, a hard solution or if it is an open problem. These notes are a modified version of similar lectures notes in Portuguese that I have used at IST-Lisbon. For the Portuguese version I have profited from comments from Ana Rita Pires, Georgios Kydonakis, Miguel Negr˜ao, Miguel Olmos, Ricardo Inglˆes, Ricardo Joel, Jos´eNat´ario and Roger Picken. A special thanks goes to my colleague from IST Silvia Anjos, who has pointed out many typos and mistakes, and has suggested several corrections. I continue to update these notes and I will be grateful for any corrections and suggestions for improvement that are sent to me. Rui Loja Fernandes [email protected] Department of Mathematics, UIUC Urbana IL, USA 3 4 Part 1. Basic Concepts The notion of a smooth manifold of dimension d makes precise the con- cept of a space which locally looks like the usual euclidean space Rd. Hence, it generalizes the usual notions of curve (locally looks like R1) and surface (locally looks like R2). This course consists of a precise study of this fun- damental concept of Mathematics and some of the constructions associated with it. We will see that many constructions familiar in infinitesimal anal- ysis (i.e., calculus) extend from euclidean space to smooth manifolds. On the other hand, the global analysis of manifolds requires new techniques and methods, and often elementary questions lead to open problems. In this first part of the lectures we will introduce the most basic concepts of Differential Geometry, starting with the precise notion of a smooth manifold. The main concepts and ideas to keep in mind from these first part are: Section 0: A manifold as a subset of Euclidean space, and the various • categories of manifolds: topological, smooth and analytic manifolds. Section 1: The abstract notion of smooth manifold (our objects) and • smooth map (our morphisms). Section 2: A technique of gluing called partitions of unity. • Section 3: Manifolds with boundary and smooth maps between man- • ifolds with boundary. Section 4: Tangent vector, tangent space (our infinitesimal objects). • Section 5: The differential of a smooth map (our infinitesimal mor- • phisms). Section 6: Important classes of smooth maps: immersions, submer- • sions and local difeomorphisms. Submanifolds (our sub-objects). Section 7: Embedded sub manifolds and the Whitney Embedding The- • orem, showing that any smooth manifold can be embedded in some Euclidean space Rn. Section 8 Foliations, which are certain partitions of a manifold into • submanifolds, a very useful generalization of the notion of manifold. Section 9: Quotients of manifolds, i.e., smooth manifolds obtained • from other smooth manifolds by taking equivalence relations. 5 6 0. Manifolds as subsets of Euclidean space Recall that the Euclidean space of dimension n is: Rn := (x1,...,xn) : x1,...,xn R ∈ We will also denote by xi: Rn R the i-th coordinate function in Rn. If U Rn is an open set, a map f→: U Rm is called a smooth map if all its⊂ partials derivatives of every order:→ ∂rf j (x), ∂xi1 ∂xir · · · exist and are continuous functions in U. More generally, given any subset X Rn and a map f : X Rm, where X is not necessarily an open set, we say⊂ that f is a smooth map→ if for each x X there is an open neighborhood n m ∈ U R and a smooth map F : U R such that f X U = F X U . ⊂A very basic property which we→ leave as an exercise| ∩ is that:| ∩ Proposition 0.1. Let X Rn, Y Rm and Z Rp. If f : X Y and g : Y Z are smooth maps,⊂ then g ⊂f : X Z is⊂ also a smooth→ map. → ◦ → A bijection f : X Y , where X Rn and Y Rm, with inverse 1 → ⊂ 1 ⊂ map f − : Y X, such that both f and f − are smooth, is called a diffeomorphism→ and we say that X and Y are diffeomorphic subsets. Rm Rn f Y X One would like to study properties of sets which are invariant under dif- feomorphisms, characterize classes of sets invariant under diffeomorphisms, etc. However, in this definition, the sets X and Y are just too general, and it is hopeless to try to say anything interesting about classes of such diffeo- morphic subsets. One must consider nicer subsets of Euclidean space: for example, it is desirable that the subset has at each point a tangent space and that the tangent spaces vary smoothly. Recall that a subset X Rn has an induced topology, called the relative topology, where the relative⊂ open sets are just the sets of the form X U, where U Rn is an open set. ∩ ⊂ Definition 0.2. A subset M Rn is called a smooth manifold of di- mension d if each p M has a⊂ neighborhood M U which is diffeomorphic to an open set V R∈d. ∩ ⊂ 7 The diffeomorphism φ : M U V , in this definition, is called a coor- ∩ → 1 dinate system. The inverse map φ− : V M U, which by assumption is smooth, is called a parameterization. → ∩ Rn M U U M ∩ φ Rd V We have the category of smooth manifolds where: the objects are smooth manifolds; • the morphisms are smooth maps. • The reason they form a category is because the composition of smooth maps is a smooth map and the identity is also a smooth map. Examples 0.3. 1. Any open subset U Rd is itself a smooth manifold of dimension d: the .inclusion i : U ֒ Rd gives⊂ a global defined coordinate chart → 2. If f : Rd Rm is any smooth map, its graph: → Graph(f) := (x, f(x)) : x Rd Rd+m { ∈ }⊂ is a smooth manifold of dimension d: the map x (x, f(x)) is a diffeomor- phism Rd Graph(M), so gives a global parametrization7→ of Graph(f). → Rm (x, f(x)) R d 8 3. The unit d-sphere is the subset of Rd+1 formed by all vectors of length 1: Sd := x Rd+1 : x =1 . { ∈ || || } This is a d-dimensional manifold which does not admit a global parametriza- tion. However we can cover the sphere by two coordinate systems: if we let N = (0,..., 0, 1) and S = (0,..., 0, 1) denote the north and south poles, then stereographic projection relative to− N and S give two coordinate systems π : Sd N Rd and π : Sd S Rd. Ø× N −{ } → S −{ } →ÈËÖ Å Í « d Í Ë ¬ Æ « ¬ ½ Æ ¬ « Ô ½ Æ « ¬ d ´Ôµ Æ Ê 4. The only connected manifolds of dimension 1 are the line R and the circle S1.
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