LECTURE 4: SMOOTH MAPS 1. Smooth Maps Recall that a smooth function on a smooth manifold is a function f : M ! R so −1 that for any chart f'α;Uα;Vαg, f ◦ 'α is a smooth function on Vα. More generally, we can define Definition 1.1. Let M; N be smooth manifolds. We say a map f : M ! N is smooth if for any charts f'α;Uα;Vαg of M and f β;Xβ;Yβg of N, −1 −1 β ◦ f ◦ 'α : 'α(Uα \ f (Xβ)) ! β(f(Uα) \ Xβ) is smooth. The set of all smooth maps from M to N is denoted by C1(M; N). There are several natural way to endow it a topology. We remark that any smooth map f : M ! N induces a \pull-back" map f ∗ : C1(N) ! C1(M); g 7! g ◦ f: The pull-back will play an important role in the future. Example. The inclusion map ι : Sn ,! Rn+1 is smooth, since 1 ι ◦ '−1(y1; ··· ; yn) = 2y1; ··· ; 2yn; ±(1 − jyj2) ± 1 + jyj2 is a smooth map from Rn to Rn+1. Moreover, if g is any smooth function on Rn+1, the pull-back function ι∗g is just the restriction of g to Sn. n Example. The projection map π : Rn+1 n f0g ! RP is smooth, since x1 xi−1 xi+1 xn+1 ' ◦ π(x1; ··· ; xn+1) = ; ··· ; ; ; ··· ; i xi xi xi xi −1 is smooth on π (Ui) for all i. As in the Euclidean case, we can also define diffeomorphisms between smooth manifolds. Definition 1.2. Let M; N be smooth manifolds. A map f : M ! N is a diffeomor- phism if it is smooth, bijective, and f −1 is smooth. If a diffeomorphism between M and N exists, 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: 1 2 LECTURE 4: SMOOTH MAPS • If f : M ! N and g : N ! P are smooth, so is g ◦ f. • 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) = ff : M ! M j f is a diffeomorphismg is a group, called the diffeomorphism group of M. Example. If M is a smooth manifold, then any chart ('; U; V ) gives a diffeomorphism ' : U ! V from U ⊂ M to V ⊂ Rn. Conversely, one can easily check that two atlas A = f(φα;Uα;Vα)g and B = f('β;Uβ;Vβ)g of M are equivalent if and only if the identity map Id : (M; A) ! (M; B) is a diffeomorphism. Example. We have seen that on M = R, the two atlas A = f('1(x) = x; R; R)g and 3 B = f('2(x) = x ; R; R)g define non-equivalent smooth structures. However, the map 1=3 :(R; A) ! (R; B); (x) = x is a diffeomorphism. So we still think these two smooth structures are the same, although they are not equivalent. Remark. There exists topological manifolds that admits many different (=non-diffeomorphic) smooth structures. For example, we now know • (J. Milnor and M. Kervaire) the topological 7-sphere admits exactly 28 different smooth structures. • (S. Donaldson and M. Freedman) For any n 6= 4, Rn has a unique smooth struc- ture up to diffeomorphism; but on R4 there exists uncountable many distinct smooth structures, no two of which are diffeomorphic to each other. • (J. Munkres and E. Moise) Every topological manifold of dimension no more than 3 has a unique smooth structure up to diffeomorphism. All these results are deep and are beyond the scope of this course. 2. Tangent vectors We have seen that if U; V are Euclidean open sets, and f : U ! V is smooth, then one naturally gets a linear map df : TxU ! Tf(x)V , which can be think of as a linearization of f near the point x, and is extremely useful in studying properties of f. Now suppose M; N are smooth manifolds, and f : M ! N smooth. We would like to define its differential df, again as a linear map between tangent spaces, which serves as a linearization of f. But the first question is: What is the tangent space of a smooth manifold at a point? Now M is an abstract manifold, so we don't have a simple nice \geometric picture". To define tangent vectors on smooth manifolds, let's first re-interpret tangent vec- tors in the Euclidean case. Recall that for any point a 2 Rn and any vector ~v at a, LECTURE 4: SMOOTH MAPS 3 there is a conception of directional derivative at point a in the direction ~v, which send a smooth function f defined on Rn to the quantity a d D~v f = f(a + t~v): dt t=0 a 1 n 1 n So any v gives us an operator D~v : C (R ) ! R. Observe that for any f; g 2 C (R ), a a a • (linearity) D~v (αf + βg) = αD~v f + βD~v g for any α; β 2 R. a a a • (Leibnitz law) D~v (fg) = f(a)D~v g + g(a)D~v f. Any operator D : C1(Rn) ! R satisfying these two properties is called a derivative at a. So any vector at a defines a derivative at a. The next proposition tells us that a the correspondence v D~v is one-to-one, so that we can identify the set of tangent vectors at a with the set of derivatives at a: 1 n a Proposition 2.1. Any derivative D : C (R ) ! R at a is of the form D~v for some vector ~v at a. Proof. For any f 2 C1(Rn), we have n Z 1 d X f(x) = f(a) + f(a + t(x − a))dt = f(a) + (xi − ai)h (x); dt i 0 i=1 where Z 1 @f hi(x) = i (a + t(x − a))dt: 0 @x 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 n n n X X X @f D(f) = 0 + D(xi)h (a) + (ai − ai)D(h ) = D(xi) (a): i i @xi i=0 i=0 i=0 It follows that as an operator on C1(Rn), n X @ D = D(xi) j : @xi a i=1 1 n a In other words, if we let ~v = hD(x ); ··· ;D(x )i, then D = D~v . This motivates the following definition: Definition 2.2. Let M be an n-dimensional smooth manifold. A tangent vector at a 1 point p 2 M is a R-linear map Xp : C (M) ! R satisfying the Leibnitz law (1) Xp(fg) = f(p)Xp(g) + Xp(f)g(p) for any f; g 2 C1(M). 4 LECTURE 4: SMOOTH MAPS 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 TpM, and call it the tangent space TpM to M at p.. As argued above, if f is a constant function, then Xp(f) = 0. More generally, Lemma 2.3. If f = c in a neighborhood of p, then Xp(f) = 0. Proof. Let ' be a smooth function on M that equals 1 near p, and equals 0 at points where f 6= c. (The existence of such f is guaranteed by partition of unity.) Then (f − c)' ≡ 0. So 0 = Xp((f − c)') = (f(p) − c)Xp(') + Xp(f)'(p) = Xp(f): As a consequence, we see that if f = g in a neighborhood of p, then Xp(f) = Xp(g). In other words, Xp(f) is determined by the values of f near p. So one can replace C1(M) in definition 2.2 by C1(U), where U is any open set that contains p. In other words, as linear spaces, TpM is isomorphic to TpU. Remark. There is a more geometric way to define tangent vectors: Take a chart f'; U; V g around p. Let Cp be the set of all smooth maps γ :(−"; ") ! U such that γ(0) = p. Such maps are called smooth curves on M. We define an equivalence relation on Cp by d(' ◦ γ ) d(' ◦ γ ) γ ∼ γ () α (0) = β (0): α β dt dt Then TpM = Cp= ∼. One can show that these two definitions are equivalent. 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 n m in Euclidean spaces at x 2 U is a linear map dfx : TxU = R ! Tf(x)V = R whose matrix is the Jacobian matrix of f. Similarly, we can define Definition 2.4. Let f : M ! N be a smooth map. Then for each p 2 M, the differential of f is the linear map dfp : TpM ! Tf(p)N defined by dfp(Xp)(g) = Xp(g ◦ f) 1 for all Xp 2 TpM and g 2 C (N). d Example. Let γ : R ! M be a smooth curve on M with γ(0) = p, and dt the unit tangent vector on R. Then for any f 2 C1(M), d d dγ0( )(f) = (f ◦ γ): dt dt 0 d We will callγ _ (0) := dγ0( dt ) the tangent vector of the curve γ at p = γ(0)..