Differential Topology - Computer Graphics and Geometric Modeling - page 522

Riemannian metric if it is convenient. Furthermore, a Riemannian metric on a man-
ifold induces a natural Riemannian metric on every submanifold. We shall always
assume that the submanifolds have been given that induced metric.

Real-valued functions on manifolds give rise to vector fields. Assume that a dif-
ferentiable manifold M n has a Riemannian metric <,> and let f : M Æ R be a differen-
tiable function. Let p ΠM and, using the functional approach to tangent vectors, let
X 1 ( p ), X 2 ( p ),..., X n ( p ) be an orthonormal basis of the tangent space of M at p .

Definition. The vector field s for M defined by

n

Â

() = () ( ) s p X f X p .

i i

k = 1

is called the gradient of f and is denoted by —f .

The name of the gradient of f is justified because if M = R n , then we just get the
usual gradient of a function. See Exercise 8.10.2 which also presents an alternative
characterization of —f.
We return now to the subject of orientation.

Definition. An orientation of an abstract differentiable manifold M is an orientation
of the tangent bundle of M , that is, a continuously varying choice of orientations of its
tangent spaces. A manifold is said to be orientable if it admits an orientation. An ori-
ented manifold is a pair ( M ,s), where M is a manifold and s is an orientation for M .

The notion of orientation for abstract manifolds is compatible with the earlier def-
inition for submanifolds of Euclidean space in Section 8.5. This follows from the fact
that Theorem 8.5.3 holds for abstract manifolds, namely, that an n-dimensional closed
compact connected abstract manifold is orientable if and only if its nth homology
group is isomorphic to Z (the idea behind the proof is the same).

Note on orientability. One problem when discussing orientability of manifolds is
that there are different ways to define this concept. This means that one always has
to address the issue of compatibility between the different definitions and one ends
up having to state several theorems rather than just one. The one general unifying
condition for an n-dimensional connected manifold M n to be orientable is that
H n ( M ,∂ M ) ª Z . One could make that the definition of orientability but the question
of whether other very useful ways of describing it are compatible would still be there
and therefore one would not save oneself any work.

To show that the tangent bundle of S n , n ≥ 1, is orientable.

8.10.4. Example.

Solution. We could appeal to the fact that H n ( S n ) is isomorphic to Z , but it may give
the reader a little more understanding if we prove this directly. Let us use the coor-
dinate neighborhoods ( U + ,j + ) and ( U - ,j - ) defined in Example 8.8.4 and the equiva-
lence class of vectors approach to the definition of tangent vectors. If p Œ U + « U - ,