Scalar, Vector and Tensor Harmonics on the Flat Compact Orientable Three-Manifolds

TL;DR

This paper develops harmonic basis functions for scalar, vector, and tensor fields on the six compact orientable flat three-manifolds, aiding cosmological modeling and analysis of electromagnetic and gravitational fields.

Contribution

It introduces new harmonic functions for vector and tensor fields on these manifolds, extending previous scalar-based methods for cosmological applications.

Findings

Harmonic basis functions for scalar, vector, and tensor fields are constructed.

These harmonics facilitate analysis of electromagnetic and gravitational dynamics.

The work supports cosmological models with flat, compact spatial geometries.

Abstract

Observations suggest that our universe is spatially flat on the largest observable scales. Exactly six different compact orientable three-dimensional manifolds admit flat metrics. These six manifolds are therefore the most natural choices for building cosmological models based on the present observations. This paper briefly describes these six manifolds and the harmonic basis functions previously developed for representing arbitrary scalar fields on them. The principal focus of this paper is the development of new harmonics for representing arbitrary vector and second-rank tensor fields on these manifolds. These new harmonics are designed to be useful tools for analyzing the dynamics of electromagnetic and gravitational fields on these spaces.