Higher-Order Generalized Four-Recurrent Finsler Geometry: A Theoretical Framework with Potential Applications in Biomedical Imaging

Abstract

This study develops a theoretical framework for generalized four-recurrent Finsler spaces based on higher-order Cartan horizontal covariant derivatives. The work establishes mathematical relationships among Weyl's projective curvature tensor, Cartan curvature tensors, Berwald curvature tensor, Ricci tensors, torsion tensors, and deviation tensors. Several characterization theorems and necessary and sufficient conditions for higher-order recurrence are derived, providing additional structural constraints on the behavior and stability of curvature in Finsler manifolds. Particular attention is given to the ability of Finsler geometry to represent direction-dependent and anisotropic structures, which are relevant to several biomedical engineering problems. From a biomedical perspective, the proposed mathematical framework may provide a theoretical basis for representing anisotropic structures encountered in diffusion-based medical imaging, neural fiber organization, vascular architecture, and biomechanical tissue modeling. However, the present study is theoretical and does not include patient data, clinical imaging datasets, or experimental validation. The biomedical relevance presented in this work should therefore be interpreted as a potential application framework rather than as a demonstrated clinical or diagnostic application. Future studies will be required to implement the proposed geometric formulation on real biomedical imaging datasets and evaluate its computational and clinical utility.

Keywords: Finsler Geometry; Curvature Tensors; Diffusion MRI; Biomedical Imaging; Anisotropic Modeling