1.1. Bundles, connections and characteristic classes

11

UJ is called the global angular form determined by V. Conversely, show that

any angular form uniquely determines a Hermitian connection on L.

Example 1.1.8. Consider the unit sphere

S2

C

M3

with its canonical ori-

entation as the boundary of the unit ball in

1R3.

Define the open cover

{Ua,U0} by

Ua =

S2\

{south pole}

and

Up =

S2\

{north pole}.

The we can identify in an orientation preserving fashion

UaP 9* C*.

Denote by z the complex coordinate on C*. For each n G Z denote by Ln

the complex line bundle defined by the gluing cocycle

Zi3a'-C* = Uai3-C*,

Z^Zn.

Suppose V is a connection on L defined locally by uoai up where

dz

UJQ

= —n H uja.

z

Denote by F its curvature. It is a complex valued 2-form on

S2

and thus

it can be integrated over the 2-sphere. Denote by D± the upper/lower

hemisphere. D+ is identified in an orientation preserving fashion with the

unit disk {\z\ 1} C C. We have

/ F = / du;a + / dvp = / (va- Up)

JS2 JD+ JD- JdD+

=„/

Jdl

— = 27rin.

!dD

+

z

We arrive at several amazing conclusions.

• The integral of i*V is independent of V !!!

• The integral of Fy is an integer multiple of 27ri !!!

• The line bundle Ln with n ^ O cannot admit flat connections so that the

noncommutativity of partial derivatives is present for any covariant method

of differentiation !!!

• The line bundle Ln with n ^ 0 is not isomorphic to the trivial line bundle

C which admits a flat connection !!!

Exercise 1.1.14. Prove that the line bundle L\ in the above example is

isomorphic to the universal line bundle over

CP1

=

S2.