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

ft1

(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

k

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

a

)) 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

dzap

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

S1:

Z(3a

•

Uap

—

S

.

The connection is Hermitian (i.e. compatible with the metric) if uoa G

VLl{Ua)

® u(l) = iR. Thus we can write

ua = i9ai 9a G fi (Ua).

They are related by

ep-ea = -i^s. = 4

{dd

)

where d6 denotes the angular form on

S1.

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

fi1(-P)

such that

ilv = d0, VpeP.