Page 87 - Curvature and Homology

where grad f is the vector field with the components af/hi relative
to the given coordinate system and - div grad f is the scalar

Now, in a Riemannian manifold M, the equation

may hold in a given coordinate neighborhood but it does not have an
invariant meaning over M, that is, the left hand side is not a tensor field.
A generalization of the concept of a harmonic function is immediately
suggested, namely, instead of ordinary (partial) differentiation employ
covariant differentiation. Hence, grad f is the covariant vector field DJ
and the divergence of this vector field is the scalar - Af defined by

where gj, is the metric tensor field of M and covariant derivatives afe
taken with respect to the connection canonically defined by the metric.
It follows that

or, alternatively
- A f = gij (- Pf - 3 {ik,})

Hence, Laplace's equation Af = 0 is a tensor equation and reduces to
(2.7.2) in a Euclidean space in which the ui (i = 1, ..., n) are rectangular
cartesian coordinates.
Equation (2.7.4), namely, the condition that the function f be a
harmonic function is the condition that the (n - 1)-form

18 n
be closed where ci is the skew-symmetric tensor
and G = det (gij). +he discussion of 5 2.6 together with the 'inter-
pretation' of a harmonic function as a certain closed (n - 1)-form
suggests the introduction of an operator (defined in terms of the metric)
which associates to a p-form or an (n - p)-form *or defined as follows:
Let
a = a(il...i,) duil A ... A duis. (2.7.7)