Page 72 - Curvature and Homology

54 I. RIEMANNIAN MANIFOLDS

(iv) Each point P E M has a neighborhood U such that cl(U) is isomorphic
with U x G, that is, if x E v1(U) the map x-+ (~(x), +(x)) from rl(U) + U x G
is a differentiable isomorphism with $(xg) = +(x)g, g E G.
Show that M x G is a principal fibre bundle by allowing G to act on M x G
as follows: (P,g)h = (P,gh), P E M, g, h E G.
2. The submanifold &(P) associated with each P E M is a closed submanifold
of B(M,G) differentiably isomorphic with G. It is called the fibre over P. If
M' is an open submanifold of M, show that .rr-l(Mf) is a principal fibre bundle
with base space M' and structural group G.
3. Let {u,} be an open covering of M. Show that the map n-l(Ua n Up) + G
defined by
+&xg) (+a(xg))-l = +B(x) (+a(x))-'c X E rfl(Ua n Ug)
is constant on each fibre. Denote the induced maps of Ua n Ug + G by fga.
For Ua n UB # the fs, are called the transition functions corresponding to
the covering {u,}. They have the property

4. Let {ua} be an open covering of M and fg, : Ua n Up -P G, Ua n Up #
a family of differentiable maps satisfying the above relation. Construct a principal
fibre bundle B(M,G) whose transition functions are the fga.
Hint: Define Na = Ua x G for each open set Ua of the covering {Ua} and
put N = U Na. If we take as open sets in N the open sets of the Na, N becomes
a
a differentiable manifold. Define an equivalence relation - in N in the following
way: (P,g) - (P,h), if and only if h = fga(P)g. Finally, define B as the quotient
space of N by this equivalence relation. Let wl(Ua) be an open submanifold of B
differentiably homeomorphic with Ua x G. In this way, B becomes a differenti-
able manifold and one may now check conditions (i) - (iv) above.
5. Show that the homogeneous space G/H of the Lie group G by the closed
subgroup H defines a principal fibre bundle G(G/H,H) with base space G/H
and structural group If (cf. VI. E. 1).
6. Show that the bundle of frames with group G = GL(n,R) is a principal'
fibre bundle.
7. Consider the principal fibre bundle B(M,G) and let F be a differentiable
manifold on which G acts differentiably, that is the map (g,v) -+g v from
G x F + F is differentiable. The group G can be made to act differentiably
on B x F in the following manner: (x,v) + (x,v)g = (xg,g-'v). Denote by E
the quotient space (B x F)/G; the points of E are the classes [(x,~)], x E B,
v E F. Denote by 'srg the canonical projection of B onto M. A projection Q
of E onto M is defined by ~Q[(x,v)] = wB(x). For each P E M, the fibre