Problem 1 (Intrinsic definition of tangent spaces). Suppose X is a k-dimensional manifold, p E X. We give an intrinsic definition of the tangent space ToX. Recall that a C'-curve on X is a smooth map y : I - X, where I is an open interval in R, such that y' * 0. Let C(p) := {7: 1 - X : ICR and O E I, Y(0) = p, y is a C'-curve}. We begin with the local model. Let X = U C Rk is an open set. (1) For 71 and 12 in C(p), we define 71 ~72 if and only if vi(0) = 72(0). Show this is an equivalent relation. (2) We define C(p) / ~ by modulo the equivalent relation. Show that Up : C(p)/ ~-> Rk defined by - 7'(0) is well-defined and bijective. (3) Show that we can use up to view C(p) / ~ as a vector space. Denote it by TpX.(4) Suppose Y = Rm, f : X - Y is a smooth map and f(p) = q. Show that f* : TpX - TqY, f-() =[for] is a well-defined linear map. This is the differential or tangent map. (5) Show that if f : X - X is a local diffeomorphism at p and f(p) = p, then f* : TpX -+ TpX, f-()=[for] is a linear isomorphism. Now we go to the really intrinsic setting. Suppose X is a manifold, p E X. We define C(p) be the space of linear maps T : Co (X) - R satisfying the Leibniz's rule at p: T(fg) = T(f)g(p) + T(g)f(p). (6) Show that T is a linear space, and find the identity element. (7) Suppose X C RE is an open set in (7), (8), (9). Show that for any Y E C(p) we can find a map Try EC(p): Ty(f) = (for)' (0). (8) Show that [y] - Ty is a well defined map C(p)/ ~- C(p). (9) Show that [y] - T, is an linear isomorphism between C(p) / ~+ C(p). (10) Now we suppose X is a general smooth manifold of dimension k. Show that C(p) has dimension k. Then we can define C as the tangent space TpX