LECTURE 4: SMOOTH MAPS 1. Smooth Maps Recall That a Smooth

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 {ϕα,Uα,Vα}, f ◦ ϕα 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 {ϕα,Uα,Vα} of M and {ψβ,Xβ,Yβ} of N, −1 −1 ψβ ◦ f ◦ ϕα : ϕα(Uα ∩ f (Xβ)) → ψβ(f(Uα) ∩ Xβ) is smooth.

The set of all smooth maps from M to N is denoted by C∞(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 ∗ : C∞(N) → C∞(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 ◦ ϕ−1(y1, ··· , yn) = 2y1, ··· , 2yn, ±(1 − |y|2) ± 1 + |y|2 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 \{0} → RP is smooth, since x1 xi−1 xi+1 xn+1 ϕ ◦ 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 diffeomorphism 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:

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• 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) = {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 ⊂ Rn. Conversely, one can easily check that two atlas A = {(φα,Uα,Vα)} and B = {(ϕβ,Uβ,Vβ)} 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 = {(ϕ1(x) = x, R, R)} and 3 B = {(ϕ2(x) = x , R, R)} define non-equivalent smooth structures. However, the map 1/3 ψ :(R, A) → (R, B), ψ(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 structure 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 vectors in the Euclidean case. Recall that for any point a ∈ 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 ∞ n ∞ n So any v gives us an operator D~v : C (R ) → R. Observe that for any f, g ∈ C (R ), a a a • (linearity) D~v (αf + βg) = αD~v f + βD~v g for any α, β ∈ R. a a a • (Leibnitz law) D~v (fg) = f(a)D~v g + g(a)D~v f. Any operator D : C∞(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:

∞ 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 ∈ C∞(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 C∞(Rn), n X ∂ D = D(xi) | . ∂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 ∞ point p ∈ 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 ∈ C∞(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 ϕ 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)ϕ ≡ 0. So

0 = Xp((f − c)ϕ) = (f(p) − c)Xp(ϕ) + Xp(f)ϕ(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 C∞(M) in definition 2.2 by C∞(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 {ϕ, U, V } around p. Let Cp be the set of all smooth maps γ :(−ε, ε) → U such that γ(0) = p. Such maps are called smooth curves on M. We define an equivalence relation on Cp by d(ϕ ◦ γ ) d(ϕ ◦ 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 ∈ 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 ∈ M, the differential of f is the linear map dfp : TpM → Tf(p)N defined by

dfp(Xp)(g) = Xp(g ◦ f) ∞ for all Xp ∈ TpM and g ∈ 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 ∈ C∞(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).