4

Lie Groups

By using charts we can define differentiability for functions be-

tween smooth manifolds.

Definition. Let M and N be two smooth manifolds and / : M — N

a function. Then / is called smooth (or differentiable) if for any two

charts fi: U — V and (j): U — V of M and A T respectively, the map

^o/of^tt/nr1^))-^

is a smooth (differentiable) function between Euclidean spaces.

A diffeomorphism f: M —» N is a smooth function that has an

inverse which is also smooth.

Next, we will discuss tangent vectors and vector fields. Let ^(M)

be the set of all smooth real-valued functions on a manifold M.

Definition. Let p be a point of a manifold M. A tangent vector to

M at p is a real-valued function v: J-(M) — » K. that satisfies:

(a) v(af+ bg) = av{f) + bv(g),

(b) u(/0) = i;(/)^(p) +

/ ( P M S )

(Leibniz rule) (a, 6 G R , / , ^

.F(M)).

At each point p £ M let Tp(M) be the set of all tangent vectors

to M at p. Then under the operations

(v + w)(f)=v(f) + w(f),

(av)(f) = av{f),

the set TP(M) is made into a real vector space of dimension equal to

that of M. A basis for this vector space is constructed as follows:

Take a local chart (C/, 0) of p, and let Xi (i = 1,..., n) be the

^t/i component of fi (i.e., the result of the composition of 0: U — » Mn

with the zt/l projection ^ : W1 — » R.) Then the function

sending each / £ T{M) to

: ^(M )