Differential Topology - Computer Graphics and Geometric Modeling - page 509

k

b u
v

∂
∂

 1

i
j

() = aXu = . (8.17)

i i j

j =

We see that this matches equation (8.13) and the two definitions of tangent vectors
really amount to the same thing. In fact, the correspondence

k k

∂

 Â

a ´ a e U (8.18)

j jj ,

∂ u

j

j = 1 j = 1

defines the natural isomorphism between the two vector spaces that are called the
tangent space to the manifold.

Definition. Let M k and W m be differentiable manifolds and let f : M k Æ W m be a dif-
ferentiable map. If p ΠM k , then define a map

( ) Æ ( )

()

k m

Df pM : T T W (8.19a) (

)

p f p

by

(

Df ( ( ) X

)( ) =

g X g

(

o f

)

(8.19b)

for every X ΠT p ( M k ) and g ΠF(f( p )). The map Df( p ) is called the derivative of f at p .

Just like with the previous equivalence class of vectors definition, one can show
that Df( p ) is a well-defined linear transformation.
No matter which definition of tangent vectors one uses, given a map f between
differentiable manifolds, the rank of f at a point p is the rank of its derivative Df( p ).
Let us summarize the main points that we covered in this section. We defined
abstract manifolds, tangent vectors, when a map is differentiable, and the derivative
of a map. One can show that for submanifolds of Euclidean space the notions of
tangent vectors, the derivative of a map, and the rank of a map are compatible with
those given in Section 8.4.

8.9 Vector Bundles

Bundles over a space were introduced in Section 7.4.2. The basic concept consisted
of three pieces, a total space, a projection map onto a base space, and a local trivial-
ity condition (each point of the base space had a neighborhood over which the bundle
looked like a product of the base neighborhood and another space called the fiber).
In Section 7.4.2 we concentrated on a very special case, that of covering spaces, where
the fiber was a discrete space. In this section we look at the case of where the fiber is
a vector space. Even though covering spaces are really part of the same general topic
of “fiber” bundles, for historical reasons the notation differs slightly between the two.
We shall now switch to the notation used for vector bundles. (In Section 7.4.2 we used
the expression “bundle over a space ” to emphasize that the base space was not part
of the definition of “bundle”, which it will be here.)