# Harmonic vector fields and the Hodge Laplacian operator on Finsler geometry

## TL;DR

This paper develops a Hodge theory for Finsler manifolds, defining harmonic forms and vector fields, and proves classification results based on the harmonic Ricci scalar, extending classical geometric analysis to Finsler geometry.

## Contribution

It introduces a natural definition of harmonic vector fields and proves a Hodge-type theorem for Finsler manifolds, linking harmonicity to the vanishing of the horizontal Laplacian.

## Key findings

- Harmonic vector fields are characterized by the vanishing of the horizontal Laplacian.
- A Bochner-Yano type classification theorem is established based on the harmonic Ricci scalar.
- A closed orientable Finsler manifold with positive harmonic Ricci scalar has zero Betti number.

## Abstract

We first present the natural definitions of the horizontal differential, the divergence (as an adjoint operator), and a $p$-harmonic form on a Finsler manifold. Next, we prove a Hodge-type theorem for a Finsler manifold in the sense that a horizontal $p$-form is harmonic if and only if the horizontal Laplacian vanishes. This viewpoint provides a new appropriate natural definition of harmonic vector fields in Finsler geometry. This approach leads to a Bochner-Yano type classification theorem based on the harmonic Ricci scalar. Finally, we show that a closed orientable Finsler manifold with a positive harmonic Ricci scalar has a zero Betti number.