Metric Tensor - Canonical Measure and Volume Form

Canonical Measure and Volume Form

In analogy with the case of surfaces, a metric tensor on an n-dimensional paracompact manifold M gives rise to a natural way to measure the n-dimensional volume of subsets of the manifold. The resulting natural positive Borel measure allows one to develop a theory of integrating functions on the manifold by means of the associated Lebesgue integral.

A measure can be defined, by the Riesz representation theorem, by giving a positive linear functional Λ on the space C₀(M) of compactly supported continuous functions on M. More precisely, if M is a manifold with a (pseudo-)Riemannian metric tensor g, then there is a unique positive Borel measure μ_g such that for any coordinate chart (U,φ),

for all ƒ supported in U. Here det g is the determinant of the matrix formed by the components of the metric tensor in the coordinate chart. That Λ is well-defined on functions supported in coordinate neighborhoods is justified by Jacobian change of variables. It extends to a unique positive linear functional on C₀(M) by means of a partition of unity.

If M is in addition oriented, then it is possible to define a natural volume form from the metric tensor. In a positively oriented coordinate system (x1,...,xn) the volume form is represented as

where the dxi are the coordinate differentials and the wedge ∧ denotes the exterior product in the algebra of differential forms. The volume form also gives a way to integrate functions on the manifold, and this geometric integral agrees with the integral obtained by the canonical Borel measure.