LECTURE 1. Differentiable manifolds, differentiable maps Def: topological m-manifold. Locally euclidean, Hausdorff, second-countable space. At each point we have a local chart. (U; h), where h : U ! Rm is a topological embedding (homeomorphism onto its image) and h(U) is open in Rm. Def: differentiable structure. An atlas of class Cr (r ≥ 1 or r = 1 ) for a topological m-manifold M is a collection U of local charts (U; h) satisfying: (i) The domains U of the charts in U define an open cover of M; (ii) If two domains of charts (U; h); (V; k) in U overlap (U \V 6= ;) , then the transition map: k ◦ h−1 : h(U \ V ) ! k(U \ V ) is a Cr diffeomorphism of open sets in Rm. (iii) The atlas U is maximal for property (ii). Def: differentiable map between differentiable manifolds. f : M m ! N n (continuous) is differentiable at p if for any local charts (U; h); (V; k) at p, resp. f(p) with f(U) ⊂ V the map k ◦ f ◦ h−1 = F : h(U) ! k(V ) is m differentiable at x0 = h(p) (as a map from an open subset of R to an open subset of Rn.) f : M ! N (continuous) is of class Cr if for any charts (U; h); (V; k) on M resp. N with f(U) ⊂ V , the composition k ◦ f ◦ h−1 : h(U) ! k(V ) is of class Cr (between open subsets of Rm, resp. Rn.) f : M ! N of class Cr is an immersion if, for any local charts (U; h); (V; k) as above (for M, resp. N), the composition F = k ◦ f ◦ h−1 satisfies ker[dF (x)] = f0g for all x 2 h(U), where dF (x) 2 L(Rm; Rn) is the dif- ferential of F . (In particular, m ≤ n necessarily.) f : M ! N is a diffeomorphism if it is a homeomorphism onto N and, for any p 2 M and local charts (U; h); (V; h) at p; f(p) (resp.) and F = k ◦ f ◦ h−1, the differential dF (x) 2 L(Rm;Rn), x = h(p), is an isomorphism. (In particular, m = n.) If a Cr map f : M ! N is a diffeomorphism , the inverse f −1 is of class Cr. 1 LECTURE 2. Tangent space at a point. Recall a subset M ⊂ Rn is a m-dimensional surface of class Ck in Rn (m ≤ n) if for any p 2 M we may find an open nbd. of p, W ⊂ Rn, and an immersion of class Ck, φ : U ! Rn (U ⊂ Rm open) which is a homeomorphism onto its image: φ(U) = W \ M. n For surfaces in R , the tangent space TpM at p 2 M is geometrically defined as the set of all velocity vectors α0(0) at t = 0 of C1 curves α on M with α(0) = p. m Exercise 1. Show that TpM = dφ(x0)[R ], where φ(x0) = p, and that this subspace of Rn is independent of the choice of local parametrization φ of M near p. On a manifold, we use instead the idea of tangent space as the set of `directional derivatives' of differentiable functions, at p 2 M. m r r Def. Let M be a C manifold, CM the vector space of real-valued Cr functions on M; let p 2 M.A tangent vector at p is a linear map r Xp : CM ! R satisfying: r (i)p: if f; g 2 CM coincide in an open neighborhood of p, then Xp(f) = Xp(g): (ii)p: Xp(fg) = f(p)Xp(g) + g(p)Xp(f) (Leibniz rule.) It follows easily from these properties that Xp(const) = 0;Xp(f) = Xp(f + const). (Show this.) The space TpM of tangent vectors at p is a vector space of dimension m: m it admits a standard basis (@xi jp)i=1 associated to a local chart (U; h) at p such that h(p) = 0: @(f ◦ h−1) @xi jp(f) = (0); where h(p) = 0; i = 1; : : : ; m: @xi r Indeed, if Xp 2 TpM and f 2 CM with f(p) = 0 (as we may assume), we compute using (ii)p (see notes): X i i i Xp(f) = v @xi jp(f); v = Xp(x ); i where xi : U ! R is the ith component of the Cr map h : U ! R. Def. The differential of a C1 map f : M m ! N n at p 2 M is the linear map df(p) 2 L(TpM; Tf(p)N) given by: r df(p)[Xp] = Yf(p);Yf(p)(g) = Xp(g ◦ f); g 2 CN : 2 r Note that g ◦ f 2 CM (check in a local chart.) r Exercise 2. The linear map Yf(p) 2 L(CN ; R) satisfies the conditions (i),(ii) at f(p), required of an element of Tf(p)N. Reality check. If the manifold N is R (the real line), the tangent space at t0 2 R is one-dimensional: dg @ (g) = (t ); for g 2 C1( ); T N = fc@ ; c 2 g: tjt0 dt 0 R t0 tjt0 R We have an isomorphism: dg : T N ! ; (Y ) = Y (id ) = c; where Y (g) = c (t ): It0 t0 R It0 t0 t0 R t0 dt 0 1 Now let f : M ! N = R be C . Then: 1 df(p)[Xp] = Yf(p); where Yf(p)(g) = Xp(g ◦ f); for all g 2 C (R): In particular, if g = idR, we have Yf(p)(idR) = Xp(f): Thus, under the isomorphism If(p), which identifies Tf(p)N with R, we have: If(p)(df(p)[Xp]) = (df(p)[Xp])(idR) = Yf(p)(idR) = Xp(f): Informally, this identifies Xp(f) ∼ df(p)[Xp]: Xp(f) is a `directional deriva- tive' of f, as expected. LECTURE 3. Tangent bundle; locally trivial vector bundles Def. The tangent bundle of a Cr manifold M m is the set: r TM = f(p; Xp) 2 M × L(CM ; R); Xp 2 TpMg; that is, Xp satisfies (i)p; (ii)p. We have the standard projection map π : TM ! M; π(p; (Xp)) = p. (Intuitively, we think of elements of TM as `vectors with basepoint'{only vectors with the same basepoint can be added.) We endow TM with a topology by imposing two conditions: (a) π is a continuous map: π−1(U) is open in TM if U ⊂ M is an open set; (b) Let (U; h) be a local chart for M, h(p) = 0. Define h~ : π−1(U) ! U × Rm by h~(p; Xp) = (h(p); v); 3 P i where Xp = i v @ijp, the components of Xp in the basis @ijp of TpM asso- ciated with the chart h. We require h~ to be a homeomorphism. Exercise 3. Show that (a),(b) define a topology on TM making it a topological manifold (Hausdorff, second countable, locally euclidean.) De- scribe a local basis of neighborhoods at a point (p; Xp) of TM. Def. Differentiable structure on TM. Let (U; '); (V; ) be overlapping charts for M, U \ V 6= ;. So F = ◦'−1 : '(U \V ) ! (U \V ) is a Cr diffeomorphism. Then the associated charts for TM also overlap, with the transition diffeomorphism (of class Cr−1): F~ = ~ ◦ '~−1 : '(U \ V ) × Rm ! (U \ V ) × Rm: (x; α) 7! (F (x); dF (x)[α]): Thus the Cr atlas on M induces a Cr−1 atlas on TM. The tangent bundle is a special case of an important class of manifolds. Def. A locally trivial vector bundle with typical fiber V is a triple (E; M; π), where E and M are differentiable manifolds and π : E ! M a differentiable map, satisfying: (i) Any p 2 M admits a neighborhood U ⊂ M and a diffeomorphism −1 (`local trivialization') ΦU : π (U) ! U × V commuting with the identity on U: π(e) = p1(ΦU (e)), where p1 : U × V ! U is the first projection. (ii) If two trivializing domains overlap (U \ W 6= ;), the transition map has the form: −1 ΦW ◦ ΦU : U × V ! W × V; (x; v) ! (x; T (x)v); where T : U \ W ! GL(V) is a differentiable map. E is trivial if there is a global diffeomorphism Φ : E ! M × V, π(e) = −1 p1(Φ(e)), so that ΦjEp 2 L(Ep; V), Ep = π (p). Def. A section of E is a differentiable map X : M ! E such that π ◦ σ = idM , i.e. X(p) 2 Ep.A vector field on M is a section of TM: X(p) 2 TpM. Exercise 4: E is trivial , 9 a differentiable basis of sections X1;:::;Xn (n = dim V), i.e. fXi(p)g is a basis of Ep, for all p 2 M. Example 1: TS1 is trivial: V (x; y) = (−y; x) is a nonvanishing tangent vector field onS1. 4 Example 2: TS2 is not trivial: there is no non-vanishing tangent vector field on S2. This follows from the following facts: (i) If f; g; 2 C(Sn;Sn) and f(x) + g(x) 6= 08x 2 Sn ⊂ Rn+1, then f is homotopic to g. (ii) In particular, if f 2 C(S2;S2) has no fixed points, f is homotopic to α, the antipodal map α. (iii) The identity map of S2 is not homotopic to the antipodal map (this is true for even-dimensional spheres). The proof uses the homotopy invariance of the degree of a map. (iv) Suppose V were a nonvanishing vector field on S2. Consider the x+V (x) map: f(x) = jjx+V (x)jj .