Higher-order Cartan derivatives and curvature tensor decomposition in Finsler spaces: insights into mathematical and physical applications

Abstract

This paper delves into the intricate structure of curvature tensors within the realm of Finsler geometry. By harnessing the power of higher-order Cartan derivatives, we introduce a novel decomposition scheme for curvature tensors. This innovative approach not only provides deeper insights into the geometric properties of Finsler spaces but also establishes a foundational framework for further investigations. Our findings reveal that the proposed decomposition is instrumental in unraveling the connections between curvature, torsion, and the underlying metric structure. Moreover, we demonstrate the applicability of our results to various subdomains of Finsler geometry, including Finsler information geometry and Finsler cosmology.

1. Introduction

Finsler geometry, as a generalization of Riemannian geometry, offers a flexible framework for modeling diverse physical phenomena characterized by anisotropic and position-dependent metrics. Central to the study of Finsler geometry are curvature tensors, which encapsulate the intrinsic curvature properties of the underlying space. While significant progress has been made in understanding curvature tensors in Riemannian geometry, their counterparts in Finsler geometry exhibit a richer and more complex structure. Traditional approaches to analyzing curvature tensors in Finsler geometry often rely on the concept of Cartan connection. However, these methods can become cumbersome when dealing with higher-order geometric quantities. In this paper, we propose a novel approach that leverages the power of higher-order Cartan derivatives to systematically decompose curvature tensors.

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Al-Qashbari, A.M., AL-ssallal, F.A.M. & Haoues, M. (2026). Higher-order Cartan derivatives and curvature tensor decomposition in Finsler spaces: insights into mathematical and physical applications. Asia Mathematika, 10(1), 73–86. https://doi.org/10.5281/zenodo.20997283

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