LECTURE 5: SMOOTH MAPS 1. Smooth Maps Recall that a smooth function on a smooth manifold M is a function f : M R so that for any chart 1 {ϕ α, U α, V α } of M, the function f ϕ 1 α is a smooth function on V α. More generally, we can define smooth maps between smooth manifolds: Definition 1.1. Let M, N be smooth manifolds. We say a continuous map f : M N is smooth if for any chart {ϕ α, U α, V α } of M and {ψ β, X β, Y β } of N, the map ψ β f ϕ 1 α : ϕ α (U α f 1 (X β )) ψ β (X β ) is smooth. [Note: Both ϕ α (U α f 1 (X β )) and ψ β (X β ) are Euclidian open sets.] Remark. In the definition we assume that the map f is already a continuous map. In general the smoothness of all ψ β f ϕ 1 α s does not imply the continuity of f. See Problem Set 2. Remark. One can check that if f : (M, A) (N, B) is smooth, A 1 is a chart on M that is compatible with A, and B 1 is a chart on N that is compatible with B, then f : (M, A 1 ) (N, B 1 ) is smooth. Remark. A map f = (f 1,, f n ) : M R k is smooth if and only if each f i C (M). The set of all smooth maps from M to N is denoted by C (M, N). We will leave it as an exercise to prove that if f C (M, N) and g C (N, P ), then g f C (M, P ). As a consequence, any smooth map f : M N induces a pullback map f : C (N) C (M), g g f. The pull-back will play an important role in the future. 1 In this course, when we say any chart of a smooth manifold M, we always mean any chart in a given atlas A that defines the smooth structure of M. 1
2 LECTURE 5: SMOOTH MAPS Example. The inclusion map ι : S n R n+1 is smooth, since ι ϕ 1 ± (y 1,, y n ) = 1 ( 2y 1,, 2y n, ±(1 y 2 ) ) 1 + y 2 are smooth maps from R n to R n+1. Moreover, if g is any smooth function on R n+1, the pull-back function ι g is ust the restriction of g to S n. Example. The proection map π : R n+1 \ {0} RP n is smooth, since ( ) x ϕ i π(x 1,, x n+1 1 ) = x,, xi 1, xi+1,, xn+1 i x i x i x i is smooth on π 1 (U i ) = {(x 1,, x n+1 ) : x i 0} for each i. As in the Euclidean case, we can define diffeomorphisms between smooth manifolds. Definition 1.2. Let M, N be smooth manifolds. A map f : M N is a diffeomorphism if it is smooth, biective, and f 1 is smooth. If there exists a diffeomorphism f : M N, then we say M and N are diffeomorphic. Sometimes we will denote M N. We will regard diffeomorphic smooth manifolds as the same. The following properties can be easily deduced from the corresponding properties in the Euclidean case: The identity map Id : M M is a diffeomorphism. If f : M N and g : N P are diffeomorphisms, so is g f. If f : M N is a diffeomorphism, so is f 1. Moreover, dim M = dim N. So in particular, for any smooth manifold M, Diff(M) = {f : M M f is a diffeomorphism} is a group, called the diffeomorphism group of M. Example. If M is a smooth manifold, then any chart (ϕ, U, V ) gives a diffeomorphism ϕ : U V from U M to V R n. Example. We have seen that on M = R, the two atlas A = {(ϕ 1 (x) = x, R, R)} and B = {(ϕ 2 (x) = x 3, R, R)} define non-equivalent smooth structures. However, the map ψ : (R, A) (R, B), ψ(x) = x 1/3 is a diffeomorphism. So we still think these two smooth structures are the same, although they are not equivalent. Remark. Here are some deep results on the smooth structures: (J. Milnor and M. Kervaire) The topological 7-sphere admits exactly 28 different smooth structures. (S. Donaldson and M. Freedman) For any n 4, R n has a unique smooth structure up to diffeomorphism; but on R 4 there exist uncountable many distinct pairwise non-diffeomorphic smooth structures.
LECTURE 5: SMOOTH MAPS 3 2. The differential of a smooth map We have seen that if U, V are Euclidean open sets, and f : U V is smooth, then for any a U, one naturally gets a linear map df a : T a U T f(a) V, which can be think of as a linearization of f near the point a, and is extremely useful in studying properties of f near a. Now suppose M, N are smooth manifolds, and f : M N smooth. We would like to define its differential df p at p, again as a linear map between tangent spaces, which serves as a linearization of f near p. But the first question is: What is the tangent space of a smooth manifold at a point? Since M is an abstract manifold, we don t have a simple nice geometric picture. To define tangent vectors on smooth manifolds, let s first re-interpret (algebraically) the tangent vectors in the Euclidean case. Recall that for any point a R n and any vector v at a, there is a conception of directional derivative at point a in the direction v, which send a smooth function f defined on R n to the quantity D a vf = lim t 0 f(a + t v) f(a) t = d dt f(a + t v). t=0 So any v gives us an operator D a v : C (R n ) R. In coordinates, if v = v 1,, v n, then D a vf = v i f x i, in other words, as an operator on C (R n ), Observe that for any f, g C (R n ), D v a = v i x. i (linearity) D v a(αf + βg) = αda v f + βda v g for any α, β R. (Leibnitz law) D v a(fg) = f(a)da v g + g(a)da v f. In general, we can define Definition 2.1. Any operator D a : C (R n ) R satisfying these two properties is called a derivative at a. So any vector v at a defines a derivative D v a at a. It is not hard to see that the correspondence v D v a preserves linearity, i.e. Da α v+β w = αda v + βda w. The next proposition tells us that the correspondence v D v a is one-to-one, so that we can identify the set (vector space) of tangent vectors at a with the set of derivatives at a: Proposition 2.2. Any derivative D : C (R n ) R at a is of the form D v a for some vector v at a. Proof. For any f C (R n ), we have f(x) = f(a) + 1 0 d dt f(a + t(x a))dt = f(a) + (x i a i )h i (x), i=1
4 LECTURE 5: SMOOTH MAPS where 1 f h i (x) = (a + t(x a))dt. 0 xi Note that the Leibnitz property implies D(1) = 0 since D(1) = D(1 1) = 2D(1). By linearity, D(c) = 0 for any constant c. So D(f) = 0 + D(x i )h i (a) + (a i a i )D(h i ) = It follows that as an operator on C (R n ), D = D(x i ) x i. x=a In other words, if we let v = D(x 1 ),, D(x n ), then D = D a v. i=1 D(x i ) f x i (a). This motivates the following definition: Definition 2.3. Let M be an n-dimensional smooth manifold. A tangent vector at a point p M is a R-linear map X p : C (M) R satisfying the Leibnitz law (1) X p (fg) = f(p)x p (g) + X p (f)g(p) for any f, g C (M). It is easy to see that the set of all tangent vectors of M at p is a linear space. We will denote this set by T p M, and call it the tangent space T p M to M at p.. As argued above, if f is a constant function, then X p (f) = 0. More generally, Lemma 2.4. If f = c in a neighborhood of p, then X p (f) = 0. Proof. Let ϕ be a smooth function on M that equals 1 near p, and equals 0 at points where f c. (The existence of such f is guaranteed by partition of unity.) Then (f c)ϕ 0. So 0 = X p ((f c)ϕ) = (f(p) c)x p (ϕ) + X p (f)ϕ(p) = X p (f). As a consequence, we see that if f = g in a neighborhood of p, then X p (f) = X p (g). In other words, X p (f) is determined by the values of f near p. So one can replace C (M) in Definition 2.3 by C (U), where U is any open set that contains p. In other words, as linear spaces, T p M is isomorphic to T p U. Finally we are ready to define the differential of a smooth map between smooth manifolds. Recall that the differential of a smooth map f : U V between open sets in Euclidean spaces at a U is a linear map df a : T a U = R n x T f(a) V = R m y whose matrix is the Jacobian matrix ( f i ) of f at a. To transplant this conception to x smooth maps, we need to take a closer look at the two interpretation of T a U: We have
LECTURE 5: SMOOTH MAPS 5 seen that we can identify the (geometric )vector v at a with the (algebraic) derivative D v a = v i. Note that geometrically, x i ( ) fi f 1 df a ( v) = v = x x v,, f n x v. The vector in the right hand side is a vector in R m y. When interpreted as a derivative on C (R m y ), it is a map that maps g C (R m y ) to v f i g x y = v i x (g f) = Da v(g f). i In other words, the derivative that corresponds to the vector df a ( v) is the derivative at f(a) that maps g C (R m ) to D v a (g f). Motivated by these computations, we define Definition 2.5. Let f : M N be a smooth map. Then for each p M, the differential of f is the linear map df p : T p M T f(p) N defined by for all X p T p M and g C (N). df p (X p )(g) = X p (g f) The chain rule still holds for composition of smooth maps: Theorem 2.6 (Chain rule). Suppose f : M N and g : N P are smooth maps, then d(g f) p = dg f(p) df p. Proof. For any X p T p M and h C (P ), d(g f) p (X p )(h) = X p (h g f) = df p (X p )(h g) = dg f(p) (df p (X p ))(h). So the theorem follows. Obviously the differential of the identity map is the identity map between tangent spaces. By repeating the proof of Theorem 1.2 in Lecture 2 we get Corollary 2.7. If f : M N is a diffeomorphism, then df p : T p M T f(p) N is a linear isomorphism. In particular, we have Corollary 2.8. If dim M = n, then T p M is an n-dimensional linear space. Proof. Let {ϕ, U, V } be a chart near p. Then ϕ : U V is a diffeomorphism. It follows that dim T p M = dim T p U = dim T f(p) V = n. In particular, we see that the tangent vectors i := dϕ 1 ( x i ) form a basis of T p M. In coordinates, one has the following explicit formula for i : i : C (U) R, i (f) = (f ϕ 1 ) (ϕ(p)). x i