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
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