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 )
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