Compatible and Almost Compatible Pseudo-Riemannian Metrics

TL;DR

This paper introduces and studies the concepts of compatible and almost compatible pseudo-Riemannian metrics, extending the idea of flat pencils of metrics and motivated by Poisson structures in hydrodynamics.

Contribution

It defines and analyzes the properties of compatible and almost compatible pseudo-Riemannian metrics, generalizing existing notions in differential geometry.

Findings

Introduced new definitions for compatible and almost compatible metrics.

Established foundational properties and relations between these metrics.

Extended the framework of flat pencils of metrics to broader contexts.

Abstract

Notions of compatible and almost compatible pseudo-Riemannian metrics, which are motivated by the theory of compatible (local and nonlocal) Poisson structures of hydrodynamic type and generalize the notion of flat pencil of metrics, are introduced and studied.