On projective Ricci curvature of cubic metrics

Yanlin Li; Yuquan Xie; Manish Kumar Gupta; Suman Sharma; Yanlin Li; Yuquan Xie; Manish Kumar Gupta; Suman Sharma

doi:10.3934/math.2025513

AIMS Mathematics

2025, Volume 10, Issue 5: 11305-11315. doi: 10.3934/math.2025513

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Research article

On projective Ricci curvature of cubic metrics

1.
School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China
2.
Department of Mathematics, Guru Ghasidas Vishwavidyalaya, Bilaspur 495009, India

Received: 09 March 2025 Revised: 15 April 2025 Accepted: 12 May 2025 Published: 19 May 2025
MSC : 53B40

To study the projective Ricci curvature (PRic-curvature) in Finsler geometry is interesting because it reflects the geometric properties that are invariant under the projective transformation. In this paper, we firstly derived an expression of S-curvature for the cubic Finsler metric and proved that this S-curvature vanishes if and only if $ \beta $ is a constant Killing form. Next, we obtain an explicit expression of projective Ricci curvature for the cubic metric. We also proved that for the projective Ricci-flat Finsler space, the $ 1 $-form $ \beta $ is closed, and then the Riemannian metric of $ \alpha $ is also Ricci-flat. Finally, we show that the cubic Finsler metric is of weak projective Ricci curvature if and only if it is projectively Ricci-flat.
- Finsler space,
- cubic metrics,
- S-curvature,
- Riemann curvature,
- Ricci curvature,
- projective Ricci curvature
Citation: Yanlin Li, Yuquan Xie, Manish Kumar Gupta, Suman Sharma. On projective Ricci curvature of cubic metrics[J]. AIMS Mathematics, 2025, 10(5): 11305-11315. doi: 10.3934/math.2025513

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Abstract

To study the projective Ricci curvature (PRic-curvature) in Finsler geometry is interesting because it reflects the geometric properties that are invariant under the projective transformation. In this paper, we firstly derived an expression of S-curvature for the cubic Finsler metric and proved that this S-curvature vanishes if and only if $ \beta $ is a constant Killing form. Next, we obtain an explicit expression of projective Ricci curvature for the cubic metric. We also proved that for the projective Ricci-flat Finsler space, the $ 1 $-form $ \beta $ is closed, and then the Riemannian metric of $ \alpha $ is also Ricci-flat. Finally, we show that the cubic Finsler metric is of weak projective Ricci curvature if and only if it is projectively Ricci-flat.

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