General off-diagonal integrability of metric and nonmetric geometric flow and Finsler-Lagrange-Hamilton modified Einstein equations

TL;DR

This paper reviews recent progress in Finsler-Lagrange-Hamilton geometric flow and gravity theories, emphasizing their formulation as modifications of Einstein gravity on tangent bundles with a focus on off-diagonal integrability.

Contribution

It introduces a unified axiomatic framework for Finsler-Lagrange-Hamilton gravity theories, extending Einstein gravity with new geometric flow results and off-diagonal integrability.

Findings

Development of a geometric flow framework for Finsler and Hamiltonian structures.

Formulation of modified Einstein equations in a Finsler-Lagrange-Hamilton setting.

Identification of conditions for off-diagonal integrability in these theories.

Abstract

Over the last seventy years, many Finsler-type geometric and modified gravity theories have been elaborated. They have been formulated in terms of different classes of Finsler generating functions, metric and nonmetric structures, nonlinear and linear connections, and various sets of postulated fundamental geometric objects with corresponding nonholonomic dynamical or evolution equations. In several approaches, the resulting gravitational and matter field equations were not completely defined geometrically, or were developed only for restricted models. We present a progress report with historical remarks and a summary of new results on Finsler - Lagrange - Hamilton geometric flow and gravity theories. Such theories can be constructed in an axiomatic form on (co) tangent Lorentz bundles as nontrivial modifications of Einstein gravity.