Page 96 - Curvature and Homology

78 11. TOPOLOGY OF DIFFERENTIABLE MANIFOLDS

where

Then, the Laplace-Beltrarni operator

A = d8 + 8d
is given by

(2.12.4)
P-1

In an Euclidean space, the curvature tensor vanishes, and so if the
ul, un are rectangular coordinates, gii = St and

On the other hand, in a Riemannian manifold M, if we apply A to a
function f defined aver M, we obtain Beltrami's differential operator
of the second kind:
Af = - gfj Dj Di f
(cf. formula (2.7.3)). The operator A is therefore the usual Laplacian.

EXERCISES

A. The star operator
The following seven exercises give rise to an alternate definition of the Hodge
star operator.
1. Let V be an n-dimensional vector space over R with an inner product y:
V x V + R. If a = v, A ... A vD and /3 = w, A ... A w, are two decomposable
p-vectors, let (a, /3) = det (tp(vi, wj)). Prove that this pairing defines an inner
product on AP(V).