Page 97 - Curvature and Homology

EXERCISES 79

2. Let M be an n-dimensional Riemannian manifold with metric tensor g.
In terms of a system of local coordinates (uf), let a = a(fl,..f,l dufl A ... A duf.
and = b(il...f,l duil A ... A dui, be two (anti-symmetrized) p-forms in AYV?),
P being in the given coordinate neighborhood. Show that

where the inner product Q is defined by g.
3. Let V* = V @ ... @ V (p times) and define A* : V* Vp by

the summation being taken over all permutations of the set (I, ***,p). Deb
the map
7 : Ap(V) Ap(V9)
by
~(s ... v,,) = A*(vl @ ... @up);
A
q is an isomorphism. Furthermore, if we extend Q to an inner product on VP by
(.I @ *** @ v,, w1@ *** O .I,) = (V,,W~) *** (vp,wp>,
then P! (a,B> = (7(4,4B)>*
We have used the notation dv,w) = (v,w), v,w E V. (The correspondence
between w E V and w* E V* given by the condition

defines an isomorphism between V and V*.)
4. Show that (AP(V))* A*(V*) under the pairing

<vl A ... A v,, w: A ... A w,*) = det ((vf, w:)).
5. If the manifold M is oriented, there is a unique n-form e* in An(V;),
PE M such that (e*,e*) = 1 where e* is positive with respect to the
orientation. (Note that the metric tensor g defines an inner product on V;).
6. Define h : Ap(Vp) _t A~-~(V;) by

and let a =a(fl...fr)(8/hil) A ... A (8/hfs) be an element of AP(Vp), where
the coefficients are anti-symmetrized. Then,

and G = det(g,,).