Math 549 Geometric Structures Spring 2008
Assignment #1 (CORRECTED)
Due May 9, 2008
1. Let E→Mbe a k-dimensional (real or complex) vector bundle. Suppose there exists a global
nonvanishing section Ωof ΛkE, and give Ethe corresponding structure group G=SL(k, R)
or SL(k , C). Prove that a connection ∇on Eis a G-connection if and only if ∇Ω≡0.
2. Let E→Mbe a smooth vector bundle. Recall that a connection ∇in Eis said to be flat
if its curvature vanishes identically. Show that ∇is flat if and only if for each p∈M, there
exists a parallel frame for Ein a neighborhood of p.
3. Let E→Mbe a smooth vector bundle with structure group G, and let ∇be a G-connection
in E. Show that in a neighborhood of each p∈M, there is a local G-frame (sα) for Esuch
that ∇sα= 0 at pfor each α. [Hint: Start with any G-frame at p, and parallel translate along
radial lines in some coordinate chart centered at p. Why is the resulting frame smooth?]
4. (a) Let Gbe a Lie group, and let θbe its Maurer–Cartan form. Prove that θsatisfies the
following identity, known as the Maurer-Cartan equation:
dθ +1
2[θ, θ] = 0.
(b) If P=M×G→Mis the canonical trivial principal G-bundle over M, show that π∗
2θis
a flat connection on P, where π2is pro jection onto Gand θis the Maurer–Cartan form
of G.
5. Let Gbe a Lie group and gits Lie algebra. Let P→Mbe a smooth principal G-bundle, let
ωbe a connection on P, and let H⊆T P be its horizontal distribution. For each X∈g, let
b
Xdenote the corresponding fundamental vector field on P.
(a) For each X∈g, show that Lˆ
Xω=−ad(X)◦ω, where ad: g→gl(g) is the adjoint
representation of g, defined by ad(X)(Y) = [X, Y ]. (Recall from [ISM] that ad is the
induced Lie algebra homomorphism associated with Ad : G→GL(g).)
(b) If Yis a smooth horizontal vector field on Pand X∈g, show that [
b
X, Y ] is horizontal.
(c) For any X, Y ∈g, show that [
b
X,
b
Y] = \
[X, Y ].
(d) For any smooth k-form ηon P, define the horizontal exterior derivative of ηby
dHη(X1,...,Xk+1) = dη(πHX1,...,πHXk),
where πH:T P →His the projection onto Hwith kernel V. Prove that the curvature
form Ω of ωsatisfies Ω = dHω.
6. Let Mbe a smooth n-manifold, and let F(M) be the frame bundle of M, considered as a
principal GL(n, R)-bundle with projection π:F(M)→M. A point f∈F(M) is a basis for
the tangent space TxM, where x=π(f)∈M; such a basis can be identified with a linear
isomorphism f:Rn→TxM. Define an Rn-valued 1-form θon F(M), called the soldering
form, by
θf(X) = f−1(dπf(X)).
(a) Show that θis smooth.
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