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Finite Volume Einstein Finsler Warped Product Manifolds of Non-positive or Non-negative Scalar Curvature

Mohammad Aqib, Hemangi Madhusudan Shah, Pankaj Kumar, Anjali Shriwastawa

TL;DR

The paper investigates the existence and rigidity of Einstein warped product manifolds in Finsler geometry, focusing on bases that are compact or finite-volume Riemannian and fibers that are weakly Berwald. By deriving exact Ricci and scalar-curvature relations for Finsler warped products and analyzing volume-finiteness under various Finsler volume forms, it extends classical Riemannian results (Besse, Kim–Kim, Dumitru) to the Finsler setting, and provides both obstruction theorems and existence results for finite-volume Einstein warped products. A key methodological pillar is the use of the warp function’s Hessian and the Omori–Yau maximum principle to obtain rigidity under nonpositive curvature, leading to trivial (product) structures in many cases, while section 6 catalogs conditions under which nontrivial examples could arise. Together, these results deepen the understanding of how curvature constraints, volume finiteness, and Finsler-type fiber geometry interact in warped-product constructions with potential applications to geometric models in relativity and beyond.

Abstract

The notion of warped product plays an important role in Riemannian geometry moreover in geodesic metric spaces. The warped product was first introduced by Bishop and O'Neill to study Riemannian manifolds of negative curvature.Warped products have been mainly used to construct new examples of Riemannian manifolds with prescribed curvature conditions. This construction can be extended for Finslerian metrics with some minor restrictions. This is motivated by Asanov's papers, where some models of relativity theory are described through the warped product of Finsler metrics. These metrics are in the form of $(α,β)$-metrics, which are the generalization of the Randers metrics; which are being asymmetric Finsler metrics in four-dimensional space-time. The product was later extended to the warped product case of Finsler manifolds by the work of Kozma, Peter and Verge.

Finite Volume Einstein Finsler Warped Product Manifolds of Non-positive or Non-negative Scalar Curvature

TL;DR

The paper investigates the existence and rigidity of Einstein warped product manifolds in Finsler geometry, focusing on bases that are compact or finite-volume Riemannian and fibers that are weakly Berwald. By deriving exact Ricci and scalar-curvature relations for Finsler warped products and analyzing volume-finiteness under various Finsler volume forms, it extends classical Riemannian results (Besse, Kim–Kim, Dumitru) to the Finsler setting, and provides both obstruction theorems and existence results for finite-volume Einstein warped products. A key methodological pillar is the use of the warp function’s Hessian and the Omori–Yau maximum principle to obtain rigidity under nonpositive curvature, leading to trivial (product) structures in many cases, while section 6 catalogs conditions under which nontrivial examples could arise. Together, these results deepen the understanding of how curvature constraints, volume finiteness, and Finsler-type fiber geometry interact in warped-product constructions with potential applications to geometric models in relativity and beyond.

Abstract

The notion of warped product plays an important role in Riemannian geometry moreover in geodesic metric spaces. The warped product was first introduced by Bishop and O'Neill to study Riemannian manifolds of negative curvature.Warped products have been mainly used to construct new examples of Riemannian manifolds with prescribed curvature conditions. This construction can be extended for Finslerian metrics with some minor restrictions. This is motivated by Asanov's papers, where some models of relativity theory are described through the warped product of Finsler metrics. These metrics are in the form of -metrics, which are the generalization of the Randers metrics; which are being asymmetric Finsler metrics in four-dimensional space-time. The product was later extended to the warped product case of Finsler manifolds by the work of Kozma, Peter and Verge.
Paper Structure (9 sections, 23 theorems, 121 equations)

This paper contains 9 sections, 23 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Admissible class of Einstein Finsler warped product manifolds of finite volume
  4. Einstein Finslerian warped product manifolds
  5. Finsler warped product manifolds of finite volume
  6. Existence of Einstein Finsler warped product of finite volume
  7. Warped product of weakly Berwald Finsler manifolds
  8. Einstein Finsler warped product manifold with Ricci a non-positive scalar function
  9. Einstein Finsler warped product manifold with Ricci a positive scalar function

Key Result

Theorem 1

If $(M, g)$ is an Einstein Riemannian warped product manifold with non-positive scalar curvature and compact base, then $M$ is simply a Riemannian product space.

Theorems & Definitions (56)

  • Theorem 1: Theorem 1 kim
  • Definition 1: Finsler metric SSZ
  • Definition 2: Finsler geodesic SSZ
  • Definition 3: Geodesic equation, Spray coefficients SSZ
  • Definition 4: The Riemann curvature tensor CXSZ
  • Definition 5: Flag Curvature SSZ
  • Definition 6: The Ricci curvature SSZ
  • Definition 7: Einstein manifold SSZ
  • Definition 8: Distortion, SSZ
  • Definition 9: S-Curvature SSZ
  • ...and 46 more