Classification of Three-Dimensional Left-Invariant Ricci-Quadratic Randers Metrics and its Applications

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics and Computer Sciences, Amirkabir University of Technology (Tehran Polytechnic), Tehran. Iran.

2 Department of Mathematics, Faculty of Science, University of Qom. Qom. Iran.

Abstract

‎In this paper‎, ‎we establish necessary and sufficient conditions for a left-invariant Randers metric to be Ricci-quadratic‎. ‎Using Ha-Bumlees classification of left-invariant Riemannian metrics and our characterization of left-invariant Ricci-quadratic Randers metrics‎, ‎we classify all left-invariant Ricci-quadratic Randers metrics on three-dimensional Lie groups‎. ‎As an application‎, ‎we provide a counterexample to Hu-Dengs theorem [HD]‎, ‎which asserts that a homogeneous Randers metric is Ricci-quadratic if and only if it is Berwald‎. ‎This demonstrates that Shen's rigidity theorem for R-quadratic metrics does not extend to Ricci-quadratic Finsler metrics‎. ‎Furthermore‎, ‎we show that our counterexample is a generalized Berwald metric (non-Berwaldian)‎, ‎as well as a generalized Douglas-Weyl metric that is neither Douglas‎, ‎Weyl‎, ‎nor projectively flat.

Keywords

Main Subjects

Computational Methods for Differential Equations

Articles in Press, Accepted Manuscript
Available Online from 19 July 2025

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History

  • Receive Date: 08 April 2025
  • Revise Date: 29 June 2025
  • Accept Date: 18 July 2025

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