On a class of weakly Landsberg metrics composed by a Riemannian metric and a conformal 1-form

Fangmin Dong; Benling Li; Fangmin Dong; Benling Li

doi:10.3934/math.20231398

AIMS Mathematics

2023, Volume 8, Issue 11: 27328-27346. doi: 10.3934/math.20231398

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Research article

On a class of weakly Landsberg metrics composed by a Riemannian metric and a conformal 1-form

Fangmin Dong ,
Benling Li ^,

School of Mathematics and Statistics, Ningbo University, Ningbo, Zhejiang Province 315211, China

Received: 14 July 2023 Revised: 06 September 2023 Accepted: 08 September 2023 Published: 26 September 2023
MSC : 53B40, 53C60

Without the quadratic restriction, there are many non-Riemannian geometric quantities in Finsler geometry. Among these geometric quantities, Berwald curvature, Landsberg curvature and mean Landsberg curvature are related directly to the famous "unicorn problem" in Finsler geometry. In this paper, Finsler metrics with vanishing weakly Landsberg curvature (i.e., weakly Landsberg metrics) are studied. For the general $ (\alpha, \beta) $-metrics, which are composed by a Riemannian metric $ \alpha $ and a 1-form $ \beta $, we found that if the expression of the metric function doesn't depend on the dimension $ n $, then any weakly Landsberg $ (\alpha, \beta) $-metric with a conformal 1-form must be a Landsberg metric. In the two-dimensional case, the weakly Landsberg case is equivalent to the Landsberg case. Further, we classified two-dimensional Berwald general $ (\alpha, \beta) $-metrics with a conformal 1-form.
- Finsler metric,
- weakly Landsberg metric,
- Landsberg metric,
- Berwald metric,
- general $ (\alpha, \beta) $-metric
Citation: Fangmin Dong, Benling Li. On a class of weakly Landsberg metrics composed by a Riemannian metric and a conformal 1-form[J]. AIMS Mathematics, 2023, 8(11): 27328-27346. doi: 10.3934/math.20231398

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Abstract

Without the quadratic restriction, there are many non-Riemannian geometric quantities in Finsler geometry. Among these geometric quantities, Berwald curvature, Landsberg curvature and mean Landsberg curvature are related directly to the famous "unicorn problem" in Finsler geometry. In this paper, Finsler metrics with vanishing weakly Landsberg curvature (i.e., weakly Landsberg metrics) are studied. For the general $ (\alpha, \beta) $-metrics, which are composed by a Riemannian metric $ \alpha $ and a 1-form $ \beta $, we found that if the expression of the metric function doesn't depend on the dimension $ n $, then any weakly Landsberg $ (\alpha, \beta) $-metric with a conformal 1-form must be a Landsberg metric. In the two-dimensional case, the weakly Landsberg case is equivalent to the Landsberg case. Further, we classified two-dimensional Berwald general $ (\alpha, \beta) $-metrics with a conformal 1-form.

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