Page 73 - Curvature and Homology

EXERCISES 55

TZI(P) C E of E is the set of points represented by the class [(x,v)] where x
is an arbitrary point of B satisfying wB(x) = P and v is an arbitrary point of F.
Show that E is a differentiable manifold by considering ril(U) as an open
submanifold of E which may be identified with U x F. In terms of the
differentiable structure given to E the map TE is differentiable. The manifold E
is known as the associated Jibre bundle of B with base space M, standard fie
F and structural group G. Note that E and B have the same base spaces and
structural groups.
8. Let F be an n-dimensional vector space with the fixed basis (v,, ..., v,).
The group G = GL(n,R) acts on F by g vi = vj. The tangent bundle is the
associated fibre bundle of R with F as standard fibre. Show that the tangent
bundle is the bundle of contravariant vectors of 5 1.3.
It is surprising indeed that a manifold structure can be defined on the set
of all tangent vectors, for there is no a pt.iori relation between tangent spaces
defined abstractly. Moreover, the idea of a vector varying continuously in a
vector space which itself varies is a prkwi remarkable.
9. Let M be a (connected) differentiable manifold and B its universal covering
space. By considering the action of the fundamental group -(M) on B, show
that B is a principal fibre bundle with base space M and structural group
.rr,(M). Show also that any covering space is an associated fibre bundle of B
with discrete standard fibre.

K. Riemannian metrics
1. It has been shown that a (connected) differentiable manifold M admits
a Riemannian metric (cf. 5 1.9). With respect to a Riemannian metric, a natural
metric d may be defined as follows: d(P, Q) is the greatest lower bound of the
lengths of all piecewise differentiable curves joining P and Q. A Riemannian
manifold is therefore a metric space. It is a complete metric space if the metric d
is complete (cf. fj 7.7). In this case the Riemannian metric is said to be complete.
Every differentiable manifold carries a complete Riemannian metric. If every
Riemannian metric carried by M is complete, M is compact [86]. A Riemannian
manifold is said to be complete if its metric is complete.