Nonholonomic Clifford and Finsler Structures, Non-Commutative Ricci Flows, and Mathematical Relativity

TL;DR

This paper reviews 18 years of research on geometric methods in physics, focusing on nonholonomic structures, non-commutative geometry, and their applications to gravity, quantum models, and modified theories of gravity.

Contribution

It introduces new geometric frameworks combining Clifford and Finsler structures with nonholonomic and non-commutative geometries for gravity and quantum models.

Findings

Development of nonholonomic geometric flow methods for Ricci solitons.

Application of Clifford and Finsler structures in quantum gravity models.

Formulation of modified gravity theories using non-commutative geometry.

Abstract

In this summary of Habilitation Thesis, it is outlined author's 18 years research activity on mathematical physics, geometric methods in particle physics and gravity, modifications and applications (after defending his PhD thesis in 1994). Ten most relevant publications are structured conventionally into three "strategic directions": 1) nonholonomic geometric flows evolutions and exact solutions for Ricci solitons and field equations in (modified) gravity theories; 2) geometric methods in quantization of models with nonlinear dynamics and anisotropic field interactions; 3) (non) commutative geometry, almost Kaehler and Clifford structures, Dirac operators and effective Lagrange-Hamilton and Riemann-Finsler spaces.