Randers Metrics with Special Riemann Curvature Properties

Cheng, Xinyue; Zhang, Shen

doi:10.1007/978-3-642-24888-7_6

Cited by 17 publications

(21 citation statements)

References 2 publications

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“…In particular, when c = 0, Finsler metrics with J = 0 are called weak Landsberg metrics. Many known Finsler metrics satisfy J + cF I = 0 (see [5], [6], [18]). B. Li and Z. Shen characterized weak Landsberg metrics in (α, β)-metrics and showed that there exist weak Landsberg metrics which are not Landsberg metrics in dimension greater than two ( [16]).…”

Section: Introductionmentioning

confidence: 99%

On conformally flat (\alpha,\beta)-metrics with relatively isotropic mean Landsberg curvature

Cheng¹,

Li²,

Zou³

2014

Publ. Math. Debrecen

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In this paper, we study conformally flat (α, β)-metrics in the form of F = αφ(β/α), where α is a Riemannian metric and β is a 1-form on the manifold. We prove that conformally flat weak Landsberg (α, β)-metrics must be either Riemannian metrics or locally Minkowski metrics. Further, we prove that, if φ(s) is a polynomial in s, then conformally flat (α, β)-metrics with relatively isotropic mean Landsberg curvature must also be either Riemannian metrics or locally Minkowski metrics.Mathematics Subject Classification: 53B40, 53C60.

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Section: Introductionmentioning

confidence: 99%

On conformally flat (\alpha,\beta)-metrics with relatively isotropic mean Landsberg curvature

Cheng¹,

Li²,

Zou³

2014

Publ. Math. Debrecen

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“…An n-dimensional Finsler metric F on a manifold M is said to have constant S-curvature if S(x, y) = (n + 1)cF (x, y) for some constant c. For example, the following Finsler metric F on the unit ball has constant S-curvature S = ± 1 2 (n + 1)F , F = |y| 2 − (|x| 2 |y| 2 − x, y 2 ) 1 − |x| 2 ± x, y 1 − |x| 2 ± a, y 1 + a, x , y ∈ T x R n , where a ∈ R n is a constant vector with |a| < 1 [3,16]. Randers metric of constant flag curvature (or R-quadratic [8]) is of constant S-curvature [2].…”

Section: Introductionmentioning

confidence: 99%

On Finsler Metrics of Constant S-Curvature

Mo¹,

Wang²

2013

Bulletin of the Korean Mathematical Society

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Abstract. In this paper, we study Finsler metrics of constant S-curvature. First we produce infinitely many Randers metrics with non-zero (constant) S-curvature which have vanishing H-curvature. They are counterexamples to Theorem 1.2 in [20]. Then we show that the existence of (α, β)-metrics with arbitrary constant S-curvature in each dimension which is not Randers type by extending Li-Shen' construction.

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“…The curvature properties of Randers metrics have been studied extensively (see [4,[6][7][8]). In particular, the second author has classified projectively flat Randers metrics with constant flag curvature [14].…”

Section: Introductionmentioning

confidence: 99%

On Locally Dually Flat Finsler Metrics

2010

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Locally dually flat Finsler metrics arise from information geometry. Such metrics have special geometric properties. In this paper, we characterize locally dually flat and projectively flat Finsler metrics and study a special class of Finsler metrics called Randers metrics which are expressed as the sum of a Riemannian metric and a one-form. We find some equations that characterize locally dually flat Randers metrics and classify those with isotropic S-curvature.

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Randers Metrics with Special Riemann Curvature Properties

Cited by 17 publications

References 2 publications

On conformally flat (\alpha,\beta)-metrics with relatively isotropic mean Landsberg curvature

On conformally flat (\alpha,\beta)-metrics with relatively isotropic mean Landsberg curvature

On Finsler Metrics of Constant S-Curvature

On Locally Dually Flat Finsler Metrics

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