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.
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