6

Lie Groups

defined by

dfP(v)(g) =v(gof)

for all v G TpM and g G T{N).

At each point p G M, the differential dfp is a linear function

between the tangent spaces.

The following proposition provides a useful method of computing

the differential of a function.

Proposition 1.1. Let f': M — iV be a smooth map between two

manifolds, and let p G M and v G TpM. Take any smooth curve

a: I — • M with a(0) = p and a'(0) = v. Then the differential of f at

p is given by

—

dt"

\t=o

dfp(v)

= 3l(/

O Q

0

We now come to vector fields. A vector field X on a manifold M

is a function that assigns to each point p G M a tangent vector Xp to

M at p. Thus X: M - TM with Xp G TpM. We can think of X as

a collection of arrows, one at each point of M. If X is a vector field

on M and / G T{M\ then X / denotes the real-valued function on

M given by

X/(p) - Xp(f) for all p G M .

The vector field X is called smooth if the function X / above is smooth

for all / G JF(M). We will denote by X(M) the set of all smooth

vector fields on a manifold M.

Now, the function defined above can be viewed as a map X: T(M)

— T{M) which sends / to Xf. This map has the properties of a

derivation, i.e., the following are satisfied:

X(af + bg) = aX(f) + bX(g) a, b G R,

X(/0) = X(f)g + /X((/) (Leibniz rule).

Conversely, any derivation D on ^(M) comes from a smooth vector

field. In fact, for each p G M define Xp: F(M) -* R by X

p

(/) =

D(f)(p)- This interpretation of vector fields as derivations leads to

an important operation on vector fields.