Douglas curvature in finsler geometry: A review of the notion and its applications in geometry and physics

TL;DR

This review surveys Douglas curvature in Finsler geometry, tracing its historical development, core characterizations (notably the vanishing Douglas tensor $D_j{}^i{}_{kl}$ and projectively affine sprays), and its extensive generalizations to GDW metrics and beyond. It surveys specialized metric families—especially $(\alpha,\beta)$-metrics, their generalizations, and spherically symmetric cases—along with a detailed treatment of transformations (conformal, Randers, $\beta$-changes) and their impact on the Douglas property. A comprehensive, table-driven synthesis accompanies each major theme, connecting metric forms to Douglas-type conditions, invariance properties, and related curvature notions (Berwald, Landsberg, $S$-curvature, etc.). The generalization program reveals deep connections between Douglas-type spaces and broader projective invariants, including weakly Douglas and generalized Douglas-Weyl spaces, with notable links to physics via Berwald-like models and gravitational theories. The paper concludes with directions for computational methods, cross-curvature relationships, and new metric constructions that may broaden applications in geometry and physics.

Abstract

This paper provides a comprehensive overview of the current state of research on Douglas curvature in Finsler spaces. It explores the significance, properties, and applications of Douglas curvature, and its role in understanding Finsler geometry. The paper reviews the historical development and significance of Douglas curvature, its characterizations, generalizations, and applications, and presents future research directions.

Theorems & Definitions (10)

• Theorem 2.1
• Theorem 3.1
• Theorem 3.2
• Theorem 4.1
• Theorem 4.2
• Theorem 5.1
• Theorem 5.2
• Theorem 5.3
• Theorem 5.4
• Theorem 5.5