Foliations induced by metallic structures

TL;DR

This paper explores conditions under which distributions induced by metallic pseudo-Riemannian structures are integrable and geodesically invariant, providing inequalities and criteria for preservation under metallic maps.

Contribution

It establishes necessary and sufficient conditions for integrability and invariance of distributions in metallic structures, including Chen-type inequalities and preservation criteria.

Findings

Conditions for integrability and geodesic invariance of distributions

Chen-type inequality for metallic distributions

Criteria for metallic map preservation

Abstract

We give necessary and sufficient conditions for the real distributions defined by a metallic pseudo-Riemannian structure to be integrable and geodesically invariant, in terms of associated tensor fields to the metallic structures and of adapted connections. In the integrable case, we prove a Chen-type inequality for these distributions and provide conditions for a metallic map to preserve these distributions. If the structure is metallic Norden, we describe the complex metallic distributions in the same spirit.