General off‐diagonal Integrability of Metric and Nonmetric Geometric Flow and Finsler–Lagrange–Hamilton Modified Einstein Equations

Abstract Over the last seventy years, many Finsler‐type geometric and modified gravity theories (MGTs) 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. A progress report with historical remarks and a summary of new results on Finsler–Lagrange–Hamilton (FLH) geometric flow and gravity theories is presented. Such theories can be constructed in an axiomatic form on (co)tangent Lorentz bundles as nontrivial modifications of Einstein gravity. They are characterized by nonlinear dispersion relations and may encode nonassociative and noncommutative corrections from string theory, quantum effects, or other MGTs. To generate exact and physically relevant solutions of the FLH–modified Einstein equations, the anholonomic frame and connection deformation method is developed. A proof of the general integrability of such FLH geometric flows and MGTs is provided, and new classes of generic off‐diagonal solutions determined by generating functions and effective sources that, in general, depend on all spacetime and (co)fiber coordinates are analyzed. In general, such off‐diagonal configurations do not exhibit horizon/hypersurface duality or holographic structures and thus lie outside the Bekenstein–Hawking thermodynamic paradigm. Instead, by extending G. Perelman's entropy concept to relativistic FLH geometric flows, new classes of geometric thermodynamic variables that characterize different FLH theories and their associated solution spaces are introduce and computed.