Differential operators on almost-Hermitian manifolds and harmonic forms

TL;DR

This paper explores differential operators on almost-Hermitian manifolds, examining their harmonic forms and cohomologies, and compares these with classical theories to deepen understanding of geometric structures.

Contribution

It introduces a unified approach to Hodge theory for various differential operators on almost-Hermitian manifolds, connecting them with classical cohomologies.

Findings

Hodge theory extends to almost-Hermitian contexts

Harmonic forms relate to classical cohomologies

Comparative analysis reveals structural similarities and differences

Abstract

We consider several differential operators on compact almost-complex, almost-Hermitian and almost-K\"ahler manifolds. We discuss Hodge Theory for these operators and a possible cohomological interpretation. We compare the associated spaces of harmonic forms and cohomologies with the classical de Rham, Dolbeault, Bott-Chern and Aeppli cohomologies.