1. Differentiable Manifolds 9
If / : M -• N and g : N - P are smooth, then we have T(p of) = Tgo Tf.
This is a direct consequence of (g o /)* = /* o #*, and it is the global version
of the chain rule. Furthermore we have T(IdM) — IdTM-
If / G C°°(M), then T / : TM - ^ T R ^ l x R . We define the differential
of / by df := pr2 oT/ : TM - R. Let t denote the identity function on R.
Then (Tf.Xx)(t) = Xx(t o / ) - **(/) , so we have df(Xx) = Xx(f).
1.12. Submanifolds. A subset N of a manifold M is called a submanifold
if for each x € N there is a chart (C7, u) of M such that u(U D N) =
x 0), where R
Then clearly N is itself
a manifold with (U Pi AT, u\(U fl AT)) as charts, where (£/, n) runs through all
submanifold charts as above.
1.13. Let / :
be smooth. A point x G
is called a regular value
of / if the rank of / (more exactly: the rank of its derivative) is q at each
point y of
In this case,
is a submanifold of
n — q (or empty). This is an immediate consequence of the implicit function
theorem, as follows: Let x = 0 G
Permute the coordinates
• • ,
such that the Jacobi matrix
has the left hand part invertible. Then u := (/,pr
has invertible differential at y, so (U,u) is a chart at any y G /
(0) , and
we have / oiT"
^ ) , so u(/
(0)) - u(E7) n (0 x
Constant rank theorem ([41, I 10.3.1]). Let f : W -•
be a smooth
mapping, where W is an open subset
If the derivative df(x) has
constant rank k for each x G W, then for each a EW there are charts (U, u)
of W centered at a and (V, v) of
centered at f(a) such that v o / o
u(U) — v(V) has the following form:
(xi,..., xn) *-+ (xi,... , x/c, 0,..., 0).
is a submanifold of W of dimension n — k for each b G f(W).
Proof. We will use the inverse function theorem several times. The deriva-
tive df(a) has rank k n,q; without loss we may assume that the upper left
(k x fc)-submatrix of df(a) is invertible. Moreover, let a = 0 and f(a) = 0.
We consider u: W -
u(x \ ... ,x
• • •,
(df^\lik (df^\lik \
0 IRn-fc I