Dimensional differentiation of harmonic tensors for variations of Riemannian metric

Abstract

The purpose of this paper is to find some properties of harmonic tensors defined in a domain with boundary, when the Riemannian metric undergoes an infinitesimal change. The variations of characteristic roots and Green’s tensor are obtained. The notion of abstract dimension is introduced to preserve the duality between differential and codifferential under the change of metric. An application of the abstract dimension to a physical problem is in the last paragraph.