LECTURE 4: SMOOTH MAPS
1. Smooth Maps
Recall that a smooth function on a smooth manifold is a function f : M → R so
that for any chart {ϕα , Uα , Vα }, f ◦ ϕ−1
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 {ϕα , Uα , Vα } of M and {ψβ , Xβ , Yβ } of N ,
−1
ψβ ◦ f ◦ ϕ−1
(Xβ )) → ψβ (f (Uα ) ∩ Xβ )
α : ϕα (Uα ∩ f
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 : S n ,→ Rn+1 is smooth, since
1
1
n
ι ◦ ϕ−1
2y 1 , · · · , 2y n , ±(1 − |y|2 )
± (y , · · · , y ) =
2
1 + |y|
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 S n .
Example. The projection map π : Rn+1 \ {0} → RPn is smooth, since
1
x
xi−1 xi+1
xn+1
1
n+1
ϕi ◦ π(x , · · · , x ) =
,··· , i , i ,··· , i
xi
x
x
x
is smooth on π −1 (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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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 ) = {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
B = {(ϕ2 (x) = x3 , R, R)} define non-equivalent smooth structures. However, the map
ψ : (R, A) → (R, B),
ψ(x) = x1/3
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 : Tx U → 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
d a
D~v f = f (a + t~v ).
dt
t=0
So any v gives us an operator
D~va
∞
: C (R ) → R. Observe that for any f, g ∈ C ∞ (Rn ),
n
• (linearity) D~va (αf + βg) = αD~va f + βD~va g for any α, β ∈ R.
• (Leibnitz law) D~va (f g) = f (a)D~va g + g(a)D~va 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
the correspondence v
D~va is one-to-one, so that we can identify the set of tangent
vectors at a with the set of derivatives at a:
Proposition 2.1. Any derivative D : C ∞ (Rn ) → R at a is of the form D~va for some
vector ~v at a.
Proof. For any f ∈ C ∞ (Rn ), we have
Z 1
n
X
d
f (x) = f (a) +
f (a + t(x − a))dt = f (a) +
(xi − ai )hi (x),
0 dt
i=1
where
Z
1
∂f
(a + t(x − a))dt.
i
0 ∂x
Note that the Leibnitz property implies D(1) = 0 since
hi (x) =
D(1) = D(1 · 1) = 2D(1).
By linearity, D(c) = 0 for any constant c. So
D(f ) = 0 +
n
X
i
D(x )hi (a) +
i=0
n
X
i
i
(a − a )D(hi ) =
n
X
i=0
D(xi )
i=0
∂f
(a).
∂xi
It follows that as an operator on C ∞ (Rn ),
D=
n
X
D(xi )
i=1
∂
|a .
∂xi
In other words, if we let ~v = hD(x1 ), · · · , D(xn )i, then D = D~va .
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)
for any f, g ∈ C ∞ (M ).
Xp (f g) = f (p)Xp (g) + Xp (f )g(p)
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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 Tp M , and call it the tangent space Tp M 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, Tp M is isomorphic to Tp U .
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 ◦ γα )
d(ϕ ◦ γβ )
γα ∼ γβ ⇐⇒
(0) =
(0).
dt
dt
Then Tp M = 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
in Euclidean spaces at x ∈ U is a linear map dfx : Tx U = Rn → Tf (x) V = Rm 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 : Tp M → Tf (p) N defined by
dfp (Xp )(g) = Xp (g ◦ f )
∞
for all Xp ∈ Tp M and g ∈ C (N ).
Example. Let γ : R → M be a smooth curve on M with γ(0) = p, and
tangent vector on R. Then for any f ∈ C ∞ (M ),
d
d dγ0 ( )(f ) = (f ◦ γ).
dt
dt
0
We will call γ̇(0) :=
dγ0 ( dtd )
the tangent vector of the curve γ at p = γ(0).
d
dt
the unit