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Chapters 4 (Pdf) This is page 142 Printer: Opaque this This is page 143 Printer: Opaque this CHAPTER 4 FORMS ON MANIFOLDS 4.1 Manifolds Our agenda in this chapter is to extend to manifolds the results of Chapters 2 and 3 and to formulate and prove manifold versions of two of the fundamental theorems of integral calculus: Stokes’ theorem and the divergence theorem. In this section we’ll define what we mean by the term “manifold”, however, before we do so, a word of encouragement. Having had a course in multivariable calculus, you are already familiar with manifolds, at least in their one and two dimensional emanations, as curves and surfaces in R3, i.e., a manifold is basically just an n-dimensional surface in some high dimensional Euclidean space. To make this definition precise let X be a subset of RN , Y a subset of Rn and f : X Y a continuous map. We recall → Definition 4.1.1. f is a ∞ map if for every p X, there exists a C ∈ neighborhood, U , of p in RN and a ∞ map, g : U Rn, which p C p p → coincides with f on U X. p ∩ We also recall: Theorem 4.1.2. If f : X Y is a ∞ map, there exists a neigh- → C borhood, U, of X in RN and a ∞ map, g : U Rn such that g C → coincides with f on X. (A proof of this can be found in Appendix A.) We will say that f is a diffeomorphism if it is one–one and onto and f and f −1 are both ∞ maps. In particular if Y is an open subset C of Rn, X is an example of an object which we will call a manifold. More generally, Definition 4.1.3. A subset, X, of RN is an n-dimensional manifold if, for every p X, there exists a neighborhood, V , of p in Rm, an ∈ open subset, U, in Rn, and a diffeomorphism ϕ : U X V . → ∩ Thus X is an n-dimensional manifold if, locally near every point p, X “looks like” an open subset of Rn. Some examples: 144 Chapter 4. Forms on Manifolds 1. Graphs of functions. Let U be an open subset of Rn and f : U R a ∞ function. Its graph → C Γ = (x,t) Rn+1 ; x U,t = f(x) f { ∈ ∈ } is an n-dimensional manifold in Rn+1. In fact the map ϕ : U Rn+1 , x (x,f(x)) → → is a diffeomorphism of U onto Γ . (It’s clear that ϕ is a ∞ map, f C and it is a diffeomorphism since its inverse is the map, π :Γ U, f → π(x,t)= x, which is also clearly ∞.) C 2. Graphs of mappings. More generally if f : U Rk is a ∞ → C map, its graph Γ = (x,y) Rn Rn , x U,y = f(x) f { ∈ × ∈ } is an n-dimensional manifold in Rn+k. 3. Vector spaces. Let V be an n- dimensional vector subspace of N R , and (e1,...,en) a basis of V . Then the linear map (4.1.1) ϕ : Rn V , (x ,...,x ) x e → 1 n → i i X is a diffeomorphism of Rn onto V . Hence every n-dimensional vector subspace of RN is automatically an n-dimensional submanifold of RN . Note, by the way, that if V is any n-dimensional vector space, not necessarily a subspace of RN , the map (5.1.1) gives us an iden- tification of V with Rn. This means that we can speak of subsets of V as being k-dimensional submanifolds if, via this identification, they get mapped onto k-dimensional submanifolds of Rn. (This is a trivial, but useful, observation since a lot of interesting manifolds occur “in nature” as subsets of some abstract vector space rather than explicitly as subsets of some Rn. An example is the manifold, O(n), of orthogonal n n matrices. (See example 10 below.) This × manifold occurs in nature as a submanifold of the vector space of n by n matrices.) 4. Affine subspaces of Rn. These are manifolds of the form p + V , where V is a vector subspace of RN , and p is some specified point in 4.1 Manifolds 145 RN . In other words, they are diffeomorphic copies of the manifolds in example 3 with respect to the diffeomorphism τ : RN RN , x x + p . p × → If X is an arbitrary submanifold of RN its tangent space a point, p X, is an example of a manifold of this type. (We’ll have more to ∈ say about tangent spaces in 4.2.) § 5. Product manifolds. Let Xi, i = 1, 2 be an ni-dimensional sub- Ni manifold of R . Then the Cartesian product of X1 and X2 X X = (x ,x ) ; x X 1 × 2 { 1 2 i ∈ i} N is an n-dimensional submanifold of R where n = n1 + n2 and RN = RN1 RN2 . → We will leave for you to verify this fact as an exercise. Hint: For p X , i = 1, 2, there exists a neighborhood, V , of p in RNi , an i ∈ i i i open set, U in Rni , and a diffeomorphism ϕ : U X V . Let i i → i ∩ i U = U U , V = V V and X = X X , and let ϕ : U X V 1 × 2 1 × 2 1 × 2 → ∩ be the product diffeomorphism, (ϕ(q1), ϕ2(q2)). 6. The unit n-sphere. This is the set of unit vectors in Rn+1: Sn = x Rn+1 , x2 + + x2 = 1 . { ∈ 1 · · · n+1 } To show that Sn is an n-dimensional manifold, let V be the open n+1 subset of R on which xn+1 is positive. If U is the open unit ball in Rn and f : U R is the function, f(x) = (1 (x2 + + x2 ))1/2, → − 1 · · · n then Sn V is just the graph, Γ , of f as in example 1. So, just as ∩ f in example 1, one has a diffeomorphism ϕ : U Sn V . → ∩ More generally, if p = (x1,...,xn+1) is any point on the unit sphere, then xi is non-zero for some i. If xi is positive, then letting σ be the transposition, i n +1 and f : Rn+1 Rn+1, the map ↔ σ → fσ(x1,...,xn) = (xσ(1),...,xσ(n)) one gets a diffeomorphism, fσ ϕ, of U onto a neighborhood of p in n ◦ S and if xi is negative one gets such a diffeomorphism by replacing f by f . In either case we’ve shown that for every point, p, in Sn, σ − σ there is a neighborhood of p in Sn which is diffeomorphic to U. 146 Chapter 4. Forms on Manifolds 7. The 2-torus. In calculus books this is usually described as the surface of rotation in R3 obtained by taking the unit circle centered at the point, (2, 0), in the (x1,x3) plane and rotating it about the x3-axis. However, a slightly nicer description of it is as the product manifold S1 S1 in R4. (Exercise: Reconcile these two descriptions.) × We’ll now turn to an alternative way of looking at manifolds: as solutions of systems of equations. Let U be an open subset of RN and f : U Rk a ∞ map. → C Definition 4.1.4. A point, a Rk, is a regular value of f if for ∈ every point, p f −1(a), f is a submersion at p. ∈ Note that for f to be a submersion at p, Df(p) : RN Rk has to → be onto, and hence k has to be less than or equal to N. Therefore this notion of “regular value” is interesting only if N k. ≥ Theorem 4.1.5. Let N k = n. If a is a regular value of f, the − set, X = f −1(a), is an n-dimensional manifold. Proof. Replacing f by τ f we can assume without loss of gener- −a ◦ ality that a = 0. Let p f −1(0). Since f is a submersion at p, the ∈ canonical submersion theorem (see Appendix B, Theorem 2) tells us that there exists a neighborhood, , of 0 in RN , a neighborhood, U , O 0 of p in U and a diffeomorphism, g : U such that O→ 0 (4.1.2) f g = π ◦ where π is the projection map RN = Rk Rn Rk , (x,y) x . × → → Hence π−1(0) = 0 Rn = Rn and by (5.1.1), g maps π−1(0) { }× O∩ diffeomorphically onto U f −1(0). However, π−1(0) is a neigh- 0 ∩ O∩ borhood, V , of 0 in Rn and U f −1(0) is a neighborhood of p in X, 0 ∩ and, as remarked, these two neighborhoods are diffeomorphic. Some examples: 8. The n-sphere. Let f : Rn+1 R → 4.1 Manifolds 147 be the map, (x ,...,x ) x2 + + x2 1 . 1 n+1 → 1 · · · n+1 − Then Df(x) = 2(x1,...,xn+1) so, if x = 0 f is a submersion at x. In particular f is a submersion 6 at all points, x, on the n-sphere Sn = f −1(0) so the n-sphere is an n-dimensional submanifold of Rn+1. 9. Graphs. Let g : Rn Rk be a ∞ map and as in example 2 let → C Γ = (x,y) Rn Rk , y = g(x) . f { ∈ × } We claim that Γ is an n-dimensional submanifold of Rn+k = Rn f × Rk. Proof. Let f : Rn Rk Rk × → be the map, f(x,y)= y g(x). Then − Df(x,y) = [ Dg(x) , I ] − k k where Ik is the identity map of R onto itself.
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