On the Generalized Projective Riemann Curvature in Finsler Geometry

TL;DR

This work extends projective Finsler geometry by introducing generalized projective sprays defined via a smooth, 1-homogeneous function $\rho_G$ and constructing corresponding generalized projective Riemann and Ricci curvatures. It establishes a rigorous equivalence (Theorem 1) between $GPR$-quadratic behavior and a specific Berwald-curvature condition, and derives a corollary connecting generalized projective curvature to classical Douglas invariants under a particular projective factor $P=-\frac{S}{n+1}$. The paper also provides concrete constructions using volume-form adjustments and directionally invariant functions, broadening the class of curvature invariants stable under projective transformations. These results deepen the understanding of projective invariants in Finsler spaces and offer tools for analyzing anisotropic curvature phenomena in generalized geometric or physical models.

Abstract

This paper explores the generalized projective Riemann curvature in Finsler geometry, focusing on the properties of projectively equivalent Finsler metrics and the invariance of their curvature structures under projective transformations. We extend the existing frameworks of projective Riemann and Ricci curvatures by introducing new characterizations of quadratic curvature properties, highlighting their geometric significance in the broader context of Finsler manifolds. Our results provide novel insights into curvature behavior under generalized projective sprays and contribute to a deeper understanding of intrinsic geometric invariants within projective classes of Finsler spaces.

Theorems & Definitions (25)

• Theorem 2.1
• Corollary 2.2
• Lemma 3.1
• Definition 4.1
• Proposition 4.2
• proof
• Corollary 4.3
• Definition 4.4
• Remark 4.5
• Proposition 4.6