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DIFFERENTIAL GEOMETRY Contents Preface 4 Part 1. Basic Concepts 6 DIFFERENTIAL GEOMETRY RUI LOJA FERNANDES Contents Preface 4 Part 1. Basic Concepts 6 Lecture 0. Manifolds as subsets of Euclidean space 8 Lecture 1. Abstract Manifolds 13 Lecture 2. Manifolds with Boundary 19 Lecture 3. Partitions of Unity 23 Lecture 4. Tangent Space and the Differential 27 Lecture 5. Immersions, Submersions and Submanifolds 38 Lecture 6. Embeddings and Whitney’s Theorem 46 Lecture 7. Foliations 55 Lecture 8. Quotients 63 Part 2. Lie Theory 72 Lecture 9. Vector Fields and Flows 73 Lecture 10. Lie Bracket and Lie Derivative 81 Lecture 11. Distributions and the Frobenius Theorem 86 Lecture 12. Lie Groups and Lie Algebras 90 Lecture 13. Integrations of Lie Algebras and the Exponential 97 Lecture 14. Transformation Groups 105 Part 3. Differential Forms 114 Lecture 15. Differential Forms and Tensor Fields 116 Lecture 16. Differential and Cartan Calculus 125 Lecture 17. Integration on Manifolds 132 Lecture 18. de Rham Cohomology 139 Lecture 19. The de Rham Theorem 144 Lecture 20. Homotopy Invariance and Mayer-Vietoris Sequence 151 Lecture 21. Computations in Cohomology 161 Lecture 22. The Degree and the Index 171 Part 4. Fiber Bundles 179 Lecture 23. Vector Bundles 181 Lecture 24. The Thom Class and the Euler Class 190 1 Lecture 25. Pull-backs and the Classification of Vector Bundles 197 Lecture 26. Connections and Parallel Transport 205 Lecture 27. Curvature and Holonomy 212 Lecture 28. Characteristic Classes 216 Lecture 29. Fiber Bundles 225 Lecture 30. Principal Fiber Bundles 232 2 3 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 “Lectures”. Each of these lectures should correspond approximately to 1 hour and 30 minutes of classroom time. How- ever, some lectures do include more material than others, which correspond to different rhythms in class. The exercises at the end of each lecture 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 lectures. 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 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. Since this is the first english version of these notes, they contain too many typos and mistakes. 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, 2013 4 5 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 series of 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 series of lectures are: Lecture 0: A manifold as a subset of Euclidean space, and the various • categories of manifolds: topological, smooth and analytic manifolds. Lecture 1: The abstract notion of smooth manifold (our objects) and • smooth map (our morphisms). Lecture 2: A technique of gluing called Partions of unity. • Lecture 3: Manifolds with boundary and smooth maps between man- • ifolds with boundary. Lecture 4: Tangent vector, tangent space (our infinitesimal objects) • and the differential of a smooth map (our infinitesimal morphisms). Lecture 5: Important classes of smooth maps: immersions, submer- • sions and local difeomorphisms. Submanifolds (our sub-objects). Lecture 6: Embedded sub manifolds and the Whitney Embedding • Theorem, showing that any smooth manifold can be embedded in some Euclidean space Rn. Lecture 7 Foliations, which are certain partitions of a manifold into • submanifolds, a very useful generalization of the notion of manifold. Lecture 8: Quotients of manifolds, i.e., smooth manifolds obtained • from other smooth manifolds by taking equivalence relations. 6 7 Lecture 0. Manifolds as subsets of Euclidean space Recall that the Euclidean space of dimension d 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:→ ∂i1+ +ir f 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 every x X has an open neighborhood n ∈ m U R where there exists a smooth map F : U R such that F X = f. ⊂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. ∩ ⊂ 8 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 9 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. What this statement means is that any connected manifold of dimension 1 is diffeomorphic to R or to S1.
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