Math 6520: Differentiable Manifolds I Taught by Reyer Sjamaar Notes by David Mehrle [email protected] Cornell University Fall 2016 Last updated December 11, 2016. The latest version is online here. Contents 1 Introduction................................. 5 1.1 Dimension............................... 9 1.2 Lots of Examples ........................... 10 1.3 The Smooth Category......................... 14 2 Tangent Vectors............................... 15 2.1 Derivations................................ 21 3 Submanifolds ................................ 26 3.1 Rank ................................... 31 3.2 Submersions and Immersions.................... 35 3.3 Embeddings.............................. 38 4 Vector Fields................................. 42 4.1 Independent vector fields on spheres................ 43 4.2 Flows.................................. 44 4.3 Derivations, Revisited ........................ 50 5 Intermezzo: Point-set topology of Manifolds .............. 54 5.1 Paracompactness ........................... 55 5.2 Partitions of Unity........................... 57 5.3 Some applications........................... 59 6 Lie Groups.................................. 63 6.1 Vector fields on Lie groups...................... 67 6.2 Actions of Lie groups on manifolds................. 69 6.3 Classical Lie Groups ......................... 73 6.4 Smooth maps on Vector Fields.................... 74 6.5 Lie algebras and the Lie bracket................... 77 6.6 Brackets and Flows.......................... 79 7 Vector bundles................................ 85 7.1 Variations on the notion of manifolds ............... 85 7.2 Vector Bundles............................. 87 7.3 Subbundles .............................. 93 8 Foliations................................... 95 8.1 Integrability.............................. 100 8.2 Distributions on Lie Groups. .................... 103 9 Differential Forms.............................. 103 9.1 Operations on Vector Bundles.................... 103 1 9.2 Alternating algebras ......................... 106 9.3 Differential Forms........................... 110 9.4 The de Rahm complex ........................ 113 9.5 De Rahm Cohomology........................ 117 9.6 Functoriality of HDR . 118 9.7 Other properties of HDR(M) . 119 2 Contents by Lecture Lecture 01 on August 23, 2016......................... 5 Lecture 02 on August 26, 2016......................... 8 Lecture 03 on August 29, 2016.......................... 11 Lecture 04 on August 31, 2016......................... 13 Lecture 05 on September 2, 2016........................ 17 Lecture 06 on September 7, 2016........................ 19 Lecture 07 on September 9, 2016........................ 22 Lecture 08 on September 12, 2016....................... 25 Lecture 09 on September 14, 2016....................... 28 Lecture 10 on September 16, 2016....................... 30 Lecture 11 on September 19, 2016....................... 34 Lecture 12 on September 21, 2016....................... 37 Lecture 13 on September 23, 2016....................... 40 Lecture 14 on September 26, 2016....................... 43 Lecture 15 on September 28, 2016....................... 45 Lecture 16 on September 30, 2016....................... 48 Lecture 17 on October 03, 2016.......................... 51 Lecture 18 on October 05, 2016......................... 54 Lecture 19 on October 07, 2016......................... 57 Lecture 20 on October 12, 2016......................... 59 Lecture 21 on October 14, 2016......................... 62 Lecture 22 on October 17, 2016......................... 66 Lecture 23 on October 19, 2016......................... 69 3 Lecture 24 on October 21, 2016......................... 72 Lecture 25 on October 24, 2016......................... 76 Lecture 26 on October 26, 2016......................... 79 Lecture 27 on October 28, 2016......................... 82 Lecture 28 on October 31, 2016......................... 85 Lecture 29 on November 3, 2016........................ 87 Lecture 30 on November 4, 2016........................ 90 Lecture 31 on November 7, 2016........................ 92 Lecture 32 on November 9, 2016........................ 95 Lecture 33 on November 11, 2016....................... 98 Lecture 34 on November 14, 2016....................... 100 Lecture 35 on November 16, 2016....................... 103 Lecture 36 on November 18, 2016....................... 106 Lecture 37 on November 21, 2016....................... 109 Lecture 38 on November 28, 2016........................111 Lecture 39 on November 30, 2016....................... 114 Lecture 40 on December 2, 2016........................ 117 4 Lecture 01: Introduction August 23, 2016 Administrative There is a class website. There will be no exams, only homework. The first one will be due on September 7th. There is no textbook, but there are many books that you might want to download. Among them, Smooth Manifolds by Lee. 1 Introduction Example 1.1. Some examples of manifolds: Example 1.2. Non-examples of manifolds. 5 Lecture 01: Introduction August 23, 2016 Often the non-manifolds are more interesting than the manifolds, but we have to understand the manifolds first. Here are the features of manifolds. • “Smooth,” as in differentiable infinitely many times everywhere. • “same everywhere,” “homogeneous” • There is a tangent space at every point that is a vector space. The non- examples above have points where we can’t define a tangent space, or it isn’t a vector space. Definition 1.3 (Notation). Let A be an open subset of Rn and f: A R a n function. Let α = (α1, α2, ::: , αn) 2 N be a tuple of non-negative integers (a multi-index). We say that the α-th derivative of f is ! @α1 @α2 @αn @αf := ··· f α1 α2 αn , @x1 @x2 @xn if it exists. The order of α is jαj = α1 + α2 + ::: + αn. Definition 1.4. We say that f is Cr if @αf exists and is continuous for all multi- indexes α of order ≤ r. We write Cr(A) = ff: A R j f is Crg r n for the collection of all C functions on open! subsets A of R . If f is Cr for every r, then we say that f is C . \ 1 C (A) := Cr(A). r≥0 1 Definition 1.5. A vector valued function F: A Rm with components 0 1 f1(x) ! B f2(x) C B C F(x) = B . C @ . A fm(x) r r For 0 ≤ r ≤ , we say that F is C if fi is C for all i = 1, ::: , m. Definition 1.6. Let B be open in Rm. Let F: A B be a map. We say that F is a 1 diffeomorphism if F is smooth (i.e. C ), bijective, and F-1 is smooth as well. ! Example 1.7. Smooth bijections need1 not be diffeomorphisms. f(x) = x3 is p3 smooth (polynomial), and has an inverse f-1(x) = (x), but f-1 fails to be differentiable at x = 0. 6 Lecture 01: Introduction August 23, 2016 Remark 1.8. If A, B are open in Rn and Rm, respectively, and f: A B is a diffeomorphism, then m = n. Why? Since f is smooth, we can take it’s derivative. So the Jacobi matrix of f exists; f is invertible and the derivative! of the inverse is the inverse of the derivative (follows from the chain rule, as in the corollary below), so the Jacobi matrix for f must be square. Proposition 1.9 (Chain Rule). If A, B, C are open subsets of Rn, Rm and R`, respectively, and we have Cr functions f g A - B - C, then g ◦ f is Cr and ! ! D(g ◦ f)(x) = Dg(f(x)) ◦ Df(x). Corollary 1.10. If f: A B is a diffeomorphism, with A ⊆ Rn and B ⊆ Rm, then Df(x) is invertible for all x. -1 ! Proof. Let g = f : B A. Then g ◦ f = idA, and f ◦ g = idB. So we have Dg(f(x)) ◦ Df(x) = D(g ◦ f)(x) = D(id ) = I ! A n And similarly, Df(x) ◦ Dg(f(x)) = Im And moreover, m = n. Definition 1.11. Let M be a topological space. A chart on M is a pair (U, φ) where U is an open subset of M (called the domain) and φ: U Rn is a map (called the coordinate map) with the properties ! (i) φ(U) is open in Rn, (ii) φ: U φ(U) is a homeomorphism (i.e. φ is continuous, bijective, and φ-1 : φ(U) U is also continuous). ! If x 2 U, then we say that (U, φ) is a chart at x. If x 2 U and φ(x) = 0 2 Rn, ! then we say that the chart is centered at x. Definition 1.12. M is locally Euclidean or a topological manifold if M admits a chart at every point. Example 1.13. An example of a topological manifold is the ice cream cone in R3. A chart might be projection onto the plane. But this isn’t a smooth manifold because of the singularity at the apex of the cone (it’s pointy, not smooth!). Definition 1.14. Given two charts (U, φ) and (V, ) on M, we can form the transition map -1 ◦ (φjU\V ) : φ(U \ V) (U \ V). 7 ! Lecture 02: Introduction August 26, 2016 The transition map is (by the definition of charts) necessarily a homeomor- phism with inverse -1 φ ◦ ( jU\V ) : (U \ V) φ(U \ V). It eventually becomes really dreary to write! the restriction every time, so we will abbreviate ◦ φ-1, respectively φ ◦ -1. The charts are compatible if ◦ φ-1 and φ ◦ -1 are smooth (equivalently, if either is a diffeomorphism). This is trivially true if U \ V = Æ. By Remark 1.8, if (U, φ: U Rn) and (V, : V Rm) are compatible and U \ V 6= Æ, then m = n. ! ! Definition 1.15. An atlas A on M is a collection of charts A = f(Uα, φα) j α 2 Ig with the properties S (i) α2I Uα = M (ii) every pair of charts (Uα, φα), (Uβ, φβ) is compatible, i.e. the transition map -1 φβα = φβ ◦ φα : φα(Uαβ) φβ(Uαβ) is smooth, where Uαβ = Uα \ Uβ. ! Given any atlas, we can always make it bigger. For example, to a world atlas we could add maps of each city, and then to that we could add all naval maps, etc. The set of atlases on M is partially ordered by inclusion. Definition 1.16. If A, B are atlases on a topological space M, we say that A ≤ B if A ⊆ B. A smooth structure on M is a maximal atlas. Definition 1.17. A (smooth) manifold is a pair (M, A) where A is a maximal atlas (smooth structure) on M. To emphasize: maximality of A means that if B is another atlas on M, and if A ⊆ B, then A = B.