Graphics Reference
In-Depth Information
Figure 8.31.
The tangent bundle of the
circle is trivial.
S
1
¥ R
tangent lines
S
1
(a)
(b)
R
1
. A more elegant way to prove this is to define a nonzero cross-section for the
tangent bundle and use this to define an isomorphism between the tangent bundle
and the trivial bundle.
Definition.
If the tangent bundle of a manifold is trivial, then the manifold is said
to be
parallelizable
.
8.10.3. Example.
The tangent bundle of
S
2
is not trivial. This fact follows from
Corollary 8.5.6 using the second alternate definition of the tangent bundle. More gen-
erally, since the tangent bundle of
any
even-dimensional sphere
S
n
does not admit a
cross-section, it is not trivial either.
Definition.
Let
M
be a differentiable manifold with tangent bundle t
M
. A cross-
section of t
M
is called a
vector field
of
M
. The vector space of vector fields of
M
shall
be denoted by Vect (
M
). More generally, if
A
Õ
M
, then a cross-section of t
M
|
A
is called
vector field
of
M
defined over
A
.
Given our second alternate definition of the tangent bundle of a manifold, it is
obvious that the new definition of a vector field for a manifold is equivalent to the
definition given in Section 8.5, but now we have a definition that also applies to
abstract manifolds.
Definition.
A
Riemannian metric for a differentiable manifold
is a Riemannian metric
for its tangent bundle, which is also assumed to be differentiable if the manifold is.
A
Riemannian manifold
is a differentiable manifold together with a Riemannian
metric.
By Theorem 8.9.7 we know that every differentiable manifold admits a Rie-
mannian metric, but it is easy to see that directly, because one can always imbed the
manifold in some Euclidean space and use the induced inner product on vectors in a
tangent plane to the manifold. Because imbeddings are not unique, one can also see
from this that many different Riemannian metrics can be defined for a manifold.
Note.
In future discussions involving differentiable manifolds we shall not hesitate
to assume, without any explicit statement, that they have been endowed with a
