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 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.
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