LECTURE 1. BACKGROUND FROM DIFFERENTIAL GEOMETRY 111is defined as follows. If v , w are two vectors in T xX, extend them to local vector
fields. Let Hv, Hw be the horizontal lifts in TE. If e EE, then
FA(v, w)(e) = ir(-Ae([Hv, Hw])).
In local coordinates,
(1.3)
Exercise 1.12. Check that the definition of curvature makes sense, i.e. FA is an
honest tensor which does not depend on the choice of extension to local vector
fields, and prove equation (1.3).
We will need the following facts for the computations in the third and fourth
lectures.Exercise 1.13. Choose local coordinates x^1 , ... ,xn on X. Let \Ji denote the dxi
component of \l A. Show that the dxi /\ dxJ component of F A equals the commutator
[\Ji, \lj]·
Exercise 1.14. 1. Show that one can extend the covariant derivative to a map\l A : D.k(X, E)-+ n,k+^1 (x, E)
by requiring that
\JA(a©s) = da©s+ (-lla/\ \lAs.
for s E C^00 (E) and a E D,k(X).2. Show that FA = \l~ : C^00 (E) -+ D.^2 (X, E).
1.5. Self-dual two-formsIn the four-dimensional world, there are two special vector bundles which we will
need. If X is a smooth 4-manifold with an orientation and a Riemannian metric,
the 2-forms have a decomposition into two IR^3 -bundles
A^2 T* X =A?__ EfJ A~constructed as follows. There is a map * : A^2 -+ A^2 defined by
w /\ *'T/ = g(w, ry) vol,
where g is the induced metric on A^2 and vol E D.^4 (X) is the volume form, defined
as follows: if e^1 , ... , e^4 is an oriented orthonormal basis for T* X at a point x E X ,
then vol(x) = e^1 /\ e^2 /\ e^3 /\ e^4.