LECTURE 6: LOCAL BEHAVIOR VIA THE DIFFERENTIAL

1. The Inverse function theorem

¶ The inverse function theorem.
Last time we showed that if f : M → N is a diffeomorphism, then the differential
dfp : Tp M → Tf (p) N is a linear isomorphism. As in the Euclidean case (see Lecture
4), the converse is not true in general (i.e., “f : M → N is smooth and dfp is a linear
isomorphism for every p ∈ M ” does not imply “f is a diffeomorphism”) but we still
have the following partial converse:
Theorem 1.1 (The Inverse Function Theorem). Let f : M → N be a smooth map such
that dfp : Tp M → Tf (p) N is a linear isomorphism, then there exists a neighborhood U1
of p and a neighborhood X1 of q = f (p) such that f |U1 : U1 → X1 is a diffeomorphism.
Proof. Take a chart (ϕ, U, V ) near p and a chart (ψ, X, Y ) near f (p) so that f (U ) ⊂ X
(which is always possible after shrinking U and V ). Since ϕ : U → V and ψ : X → Y
are diffeomorphisms,
d(ψ ◦ f ◦ ϕ−1 )ϕ(p) = dψq ◦ dfp ◦ dϕ−1 n
ϕ(p) : Tϕ(p) V = R → Tψ(q) Y = R
n

is a linear isomorphism. It follows from the inverse function theorem (c.f. Lecture
4) that there exist neighborhoods V1 of ϕ(p) and Y1 of ψ(q) so that ψ ◦ f ◦ ϕ−1 is a
diffeomorphism from V1 to Y1 . Take U1 = ϕ−1 (V1 ) and X1 = ψ −1 (Y1 ). Then
f = ψ −1 ◦ (ψ ◦ f ◦ ϕ−1 ) ◦ ϕ
is a diffeomorphism from U1 to X1 . 

¶ Local diffeomorphism v.s. global diffeomorphism.

Definition 1.2. We say a smooth map f : M → N is a local diffeomorphism near p, if
it maps an open neighborhood of p diffeomorphically to an open neighborhood of f (p).

Note that it is possible that a map is a local diffeomorphism everywhere, but still
fails to be diffeomorphism.
Example. Let f : S 1 → S 1 be given by f (eiθ ) = e2iθ . Then it is a local diffeomorphism
everywhere, but it is not a global diffeomorphism since it is not invertible. [Please
compare this example with the example on page 5 of Lecture 4.]

It turns out that the invertibility is the only obstruction for an “everywhere local
diffeomorphism” to be a global diffeomorphism:
Proposition 1.3. Suppose f : M → N is a local diffeomorphism near every p ∈ M .
If f is invertible, then f is a global diffeomorphism.

1
2 LECTURE 6: LOCAL BEHAVIOR VIA THE DIFFERENTIAL

Proof. We only need to show f −1 is smooth. Fix any q = f (p). The smoothness of f −1
at q depends only on the behaviour of f −1 near q. Since f is a diffeomorphism from a
neighborhood of p onto a neighborhood of q, f −1 is smooth at q. 

2. The constant rank theorem

¶ Submersion and immersion.
What if dfp is not a linear isomorphism? Note that a linear isomorphism is both
surjective and injective. It is natural to study the those smooth maps whose differential
is either surjective or injective:
Definition 2.1. Let f : M → N be a smooth map.
(1) f is a submersion at p if dfp : Tp M → Tf (p) N is surjective.
(2) f is an immersion at p if dfp : Tp M → Tf (p) N is injective.
We say f is a submersion/immersion if it is a submersion/immersion at each point.

Obviously
• If f is a submersion, then dim M ≥ dim N .
• If f is an immersion, then dim M ≤ dim N .
Example. In PSet 2-1-6, we see: the natural projection π : T M → M is a submersion.
Similarly, the “zero section” embedding ι : M → T M, p 7→ (p, 0) is an immersion.
Example. A local diffeomorphism is both a submersion and an immersion.
Example (Canonical submersion). If m ≥ n, then the projection map
π : Rm → Rn , (x1 , · · · , xm ) 7→ (x1 , · · · , xn )
is a submersion.
Example (Canonical immersion). If m ≤ n, then the inclusion map
ι : Rm ,→ Rn , (x1 , · · · , xm ) 7→ (x1 , · · · , xm , 0, · · · , 0)
is an immersion.

It turns out that any submersion/immersion locally looks like these two canonical
ones.
Theorem 2.2 (Canonical Submersion Theorem). Let f : M → N be a submersion at
p ∈ M , then m = dim M ≥ n = dim N , and there exist charts (ϕ1 , U1 , V1 ) around p
and (ψ1 , X1 , Y1 ) around q = f (p) such that
ψ1 ◦ f ◦ ϕ−1
1 = π|V1 .

Theorem 2.3 (Canonical Immersion Theorem). Let f : M → N be an immersion at

p ∈ M , then m = dim M ≤ n = dim N , and there exist charts (ϕ1 , U1 , V1 ) around p
and (ψ1 , X1 , Y1 ) around q = f (p) such that
ψ1 ◦ f ◦ ϕ−1
1 = ι|V1 .
 LECTURE 6: LOCAL BEHAVIOR VIA THE DIFFERENTIAL 3

¶ The constant rank theorem.

We will not prove the canonical submersion/immersion theorems above. Instead,
we will prove a more general theorem which has the canonical submersion/immersion
theorems as special cases. For this purpose, we define
Definition 2.4. We say a smooth map f : M → N is a constant rank map near p ∈ M
if there is a neighborhood U of p so that dfq has constant rank (i.e. there exists r ∈ N
so that rank(df )q ≡ r) for all q ∈ U .
Example. If f is a submersion/immersion at p, then it is a submersion/immersion near
p (why?), and thus is a constant rank map near p.
Example (“Canonical” constant rank map). More generally, by composing suitable
canonical submersion and canonical immersion, we get a constant rank map
π ι
Rm = Rr+m−r −→ Rr −→ Rr+n−r = Rn
which sends (x1 , · · · , xr , xr+1 , · · · , xm ) ∈ Rm to (x1 , · · · , xr , 0, · · · , 0) ∈ Rn .

We shall prove:
Theorem 2.5 (The Constant Rank Theorem). Let f : M → N be a smooth map so
that rank(df ) ≡ r near p. Then there exists charts (ϕ1 , U1 , V1 ) around p and (ψ1 , X1 , Y1 )
near f (p) such that that
ψ1 ◦ f ◦ ϕ−1 1 m 1 r
1 (x , · · · x ) = (x , · · · , x , 0, · · · , 0).

Proof. As usual we will convert the general case to the Euclidian case.
Step 1: The Euclidean case.
We first assume U ⊂ Rm is open, and f : U → Rn is a smooth map so that dfx
has constant rank r for all x ∈ U . By translation (in both Rm and Rn , which amounts
to composing f with suitable “translation diffeomorphisms” in both sides) we may
assume 0 ∈ U and f (0) = 0. Since rank(df )0 = r, by switching coordinates (again in
both Rm and Rn , which amounts to composing f with suitable “switching coordinates
diffeomorphisms” in both sides) we may assume that the upper-left r × r submatrix,
 
∂fi
,
∂xj 1≤i,j≤r
∂fi
of the Jacobian df = ( ∂x j )1≤i≤n,1≤j≤m is nonsingular at x = 0 (and thus is nonsingular

near x = 0).
∂fi ∂fi


[The idea: Since rank ∂x j 1≤i,j≤r = rank( ∂xj )1≤i≤n,1≤j≤m , we may try to take

f1 , · · · , fr as part of our coordinates, so that with respect to these new coordinates, f

will keep the first r coordinates unchanged.] Now define ϕ : U → Rm by
ϕ(x) = (f1 (x), · · · , fr (x), xr+1 , · · · , xm ).
Then ϕ(0) = 0, and the differential
∂fi


∂xj 1≤i,j≤r
∗
dϕ =
0 Idn−r
4 LECTURE 6: LOCAL BEHAVIOR VIA THE DIFFERENTIAL

is nonsingular at x = 0. By the inverse function theorem, ϕ is a local diffeomorphism

near 0, i.e., there exists neighborhood U1 of 0 in Rm and V1 of 0 in Rm such that
ϕ : U1 → V1 is a diffeomorphism. Note that by definition,
f ◦ ϕ−1 (f1 (x), · · · , fr (x), xr+1 , · · · , xm ) = f ◦ ϕ−1 (ϕ(x)) = f (x) = (f1 (x), · · · , fn (x))
i.e., locally near 0 we have
f ◦ ϕ−1 (x) = (x1 , · · · , xr , gr+1 (x), · · · , gn (x))
for some smooth functions gr+1 , · · · , gn (with gi (0) = 0). Moreover, by chain rule,
 
−1 Idr 0
dfϕ−1 (x) ◦ (dϕ )x = ∂gi

.
∗ ∂xj r+1≤i≤n,r+1≤j≤m

Crucial observation: Since (dϕ−1 )x is a linear isomorphism, “rank(dfx ) = r near 0”

implies “rank(dfϕ−1 (x) ◦ (dϕ−1 )x ) = r near 0”, and thus implies
∂gi
= 0, ∀r + 1 ≤ i ≤ n, r + 1 ≤ j ≤ m
∂xj
near 0. It follows that in a small neighborhood of 0, we have
gi (x) = gi (x1 , · · · , xr ), ∀r + 1 ≤ i ≤ n.
In other words, near 0 we have
f ◦ ϕ−1 (x) = (x1 , · · · , xr , gr+1 (x1 , · · · , xr ), · · · , gn (x1 , · · · , xr )).
It remains to kill these gi ’s. So we define
ψ(y) = y 1 , · · · , y r , y r+1 − gr+1 (y 1 , · · · , y r ), · · · , y n − gn (y 1 , · · · , y r ) .

in a small neighborhood of 0, and get

ψ ◦ f ◦ ϕ−1 (x1 , · · · , xr , xr+1 , · · · , xn ) = (x1 , · · · , xr , 0, · · · , 0).
It remains to check that ψ is a local diffeomorphism. Again this follows
 from the inverse
Idr 0
function theorem and a simple computation dψ0 = .
∗ Idn−r
Step 2: The general case.
The general case follows easily (by the standard trick): Take a coordinate neigh-
borhood (ϕ, U, V ) near p and (ψ, X, Y ) near f (p), so that f (U ) ⊂ X, and dfq has
constant rank r on U . Then ψ ◦ f ◦ ϕ−1 : V → Y has constant rank r since
d(ψ ◦ f ◦ ϕ−1 )x = dψf (ϕ−1 (x)) ◦ dfϕ−1 (x) ◦ (dϕ−1 )x
and since (dϕ−1 )x , dψf (ϕ−1 (x)) are linear isomorphisms. Now the desired conclusion
follows from the Euclidean case. 
As a consequence, we see a map is a constant rank map if and only if it can be
written, locally, as a composition j ◦s, where s is a submersion while j is an immersion.
In particular,
• If a constant rank map is surjective, then it is a submersion.
• If a constant rank map is injective, then it is an immersion.