2. Smooth manifolds: A review 5

is a tangent vector to M at p, and the set of all such tangent vectors

for i = 1,..., n is a basis for TP{M).

We now set TM — UpTp(M) the disjoint union over all points

of the tangent vectors at each point. Thus a point in this new space

consists of a pair (p, v), where p is a point of M and v is a tangent

vector to M at the point p. The set TM can be made into a manifold

of dimension 2n, called the tangent bundle of M. The map TT : TM —»

M given by 7r(p, f) = p (p G M, v G TpM) is called the canonical

projection. The manifold structure on TM is chosen so that TT is a

smooth map. For each p e M the pre-image n~l(p) is exactly the

tangent space TpM. It is called the fiber over p.

A cwn;e in a manifold M is a smooth map a: I — M, where / is

an open interval in K. There are several equivalent ways to define a

notion of a velocity vector

af(t)

of the curve a at t. Here we will adopt

the following: The velocity vector of a is the vector

af(t)

G Ta^M

defined by

a{t)f = —dT-{t)

for all / G T(M). This definition is motivated from the notion of

directional derivative in advanced calculus. Indeed, let a : / —

Mn

be

a smooth curve in

Rn

with a(0) = p. Let a(t) = (xi(£),..., xn(t)) G

Rn.

Then a'(0) = (zi(0),... ,(0)) = T; G

Mn.

Also, let / be a

smooth function defined in a neighborhood of p. Then by restricting

/ to the curve a, the directional derivative with respect to the vector

v G

Rn

is

d(foa)\

vf

dt

t=o

A curve is a special case of a map between manifolds. The notion

of the velocity vector (derivative of the curve) can be extended to

smooth functions between manifolds.

Definition. Let / : M -^ N be a smooth function. Then, for each

p £ M, the differential of / is the function

dfp:TpM-Tf{p)N