Diffeological Smoothness in Hodge Theory

TL;DR

This paper demonstrates that on a compact Riemannian manifold, the smoothness of families of exact forms ensures the smoothness of their associated primitives, extending classical Hodge theory results.

Contribution

It establishes the smooth dependence of primitives on parameters for families of exact forms within Hodge theory, a novel extension of classical results.

Findings

Families of exact forms have smooth primitives when the forms depend smoothly on parameters.

The result applies to compact, oriented, Riemannian manifolds.

Provides a framework for parameter-dependent Hodge decompositions.

Abstract

On a compact, oriented, Riemannian manifold, the Hodge decomposition theorem associates a smooth primitive to any exact smooth form omega. In this paper, we show that given a smooth family of exact smooth forms omega(t), the family of associated primitives is also a smooth family with respect to t.