Harmonic metallic structures

TL;DR

This paper introduces the concept of harmonic metallic structures on metallic pseudo-Riemannian manifolds, establishing conditions for harmonicity, preservation under harmonic maps, and exploring their properties on generalized tangent bundles.

Contribution

It defines harmonic metallic structures, proves their equivalence to $dJ=0$ on compact manifolds, and analyzes their behavior on generalized tangent bundles with new formulas.

Findings

Harmonic metallic structures are characterized by $dJ=0$ on compact manifolds.

Conditions for preservation of harmonic metallic structures under harmonic maps are provided.

A Weitzenb"ock formula and Hodge-Laplace operator expression are derived for these structures.

Abstract

The concept of harmonic metallic structure on a metallic pseudo-Riemannian manifold is introduced. In the case of compact manifolds we prove that harmonicity of a metallic structure $J$, with $J^2=pJ+qI$ and $p^2+4q\neq 0$, is equivalent to $dJ=0$. Conditions for a harmonic metallic structure to be preserved by harmonic maps are also given. Moreover, we consider harmonic metallic structures on the generalized tangent bundle, provide a Weitzenb\"{o}ck formula for the dual metallic structure and express the Hodge-Laplace operator on $TM \oplus T^*M$.