10 1. Preliminaries
Exercise 1.1.12. (a) Consider the trivial bundle F_M. Then the trivial
connection G is flat.
(b) If A G
(End (F)) then the curvature of 9 + A is
FA = dA + AAA.
Above, A is thought of as a matrix of with entries smooth 1-forms Uij. Then
dA is the matrix with entries the 2-forms duoij and A A A is a matrix whose
(i, j)-entry is the 2-form
If E is given by a gluing cocycle g@a and V is given by the collection of
1-forms Aa G fi1(End (F
)) then the above exercise shows that F is locally
described by the collection of 2-forms dAa + Aa A Aa.
Example 1.1.7. Suppose L —» M is a complex line bundle given by a gluing
cocycle zpa : Uap » C*. Then a connection on L is defined by a collection
of complex valued 1-forms LOa satisfying
U{3 = ViOa.
The curvature is given by the collection of 2-forms dua.
If L has a [/(l)-structure (i.e. is equipped with a Hermitian metric) then
the gluing maps belong to


The connection is Hermitian (i.e. compatible with the metric) if uoa G
® u(l) = iR. Thus we can write
ua = i9ai 9a G fi (Ua).
They are related by
ep-ea = -i^s. = 4
where d6 denotes the angular form on
Exercise 1.1.13. Consider a Hermitian line bundle L * M and denote by
P M the corresponding principal S^-bundle. For each p G P denote by
ip the injection
S1 3eu^p- eu
G P.
Suppose V is a Hermitian connection as in the above example. Show that
V naturally defines a 1-form u G
such that
ilv = d0, VpeP.
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