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Search: MSC category 53B40 ( Finsler spaces and generalizations (areal metrics) )

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1. CMB 2015 (vol 58 pp. 530)

Li, Benling; Shen, Zhongmin
 Ricci Curvature Tensor and Non-Riemannian Quantities There are several notions of Ricci curvature tensor in Finsler geometry and spray geometry. One of them is defined by the Hessian of the well-known Ricci curvature. In this paper we will introduce a new notion of Ricci curvature tensor and discuss its relationship with the Ricci curvature and some non-Riemannian quantities. By this Ricci curvature tensor, we shall have a better understanding on these non-Riemannian quantities. Keywords:Finsler metrics, sprays, Ricci curvature, non-Riemanian quantityCategories:53B40, 53C60

2. CMB 2014 (vol 57 pp. 765)

da Silva, Rosângela Maria; Tenenblat, Keti
 Helicoidal Minimal Surfaces in a Finsler Space of Randers Type We consider the Finsler space $(\bar{M}^3, \bar{F})$ obtained by perturbing the Euclidean metric of $\mathbb{R}^3$ by a rotation. It is the open region of $\mathbb{R}^3$ bounded by a cylinder with a Randers metric. Using the Busemann-Hausdorff volume form, we obtain the differential equation that characterizes the helicoidal minimal surfaces in $\bar{M}^3$. We prove that the helicoid is a minimal surface in $\bar{M}^3$, only if the axis of the helicoid is the axis of the cylinder. Moreover, we prove that, in the Randers space $(\bar{M}^3, \bar{F})$, the only minimal surfaces in the Bonnet family, with fixed axis $O\bar{x}^3$, are the catenoids and the helicoids. Keywords:minimal surfaces, helicoidal surfaces, Finsler space, Randers spaceCategories:53A10, 53B40

3. CMB 2012 (vol 57 pp. 209)

Zhao, Wei
 Erratum to the Paper "A Lower Bound for the Length of Closed Geodesics on a Finsler Manifold" We correct two clerical errors made in the paper "A Lower Bound for the Length of Closed Geodesics on a Finsler Manifold". Keywords:Finsler manifold, closed geodesic, injective radiusCategories:53B40, 53C22

4. CMB 2012 (vol 57 pp. 194)

Zhao, Wei
 A Lower Bound for the Length of Closed Geodesics on a Finsler Manifold In this paper, we obtain a lower bound for the length of closed geodesics on an arbitrary closed Finsler manifold. Keywords:Finsler manifold, closed geodesic, injective radiusCategories:53B40, 53C22

5. CMB 2011 (vol 56 pp. 184)

Shen, Zhongmin
 On Some Non-Riemannian Quantities in Finsler Geometry In this paper we study several non-Riemannian quantities in Finsler geometry. These non-Riemannian quantities play an important role in understanding the geometric properties of Finsler metrics. In particular, we study a new non-Riemannian quantity defined by the S-curvature. We show some relationships among the flag curvature, the S-curvature, and the new non-Riemannian quantity. Keywords:Finsler metric, S-curvature, non-Riemannian quantityCategories:53C60, 53B40

6. CMB 2011 (vol 56 pp. 615)

Sevim, Esra Sengelen; Shen, Zhongmin
 Randers Metrics of Constant Scalar Curvature Randers metrics are a special class of Finsler metrics. Every Randers metric can be expressed in terms of a Riemannian metric and a vector field via Zermelo navigation. In this paper, we show that a Randers metric has constant scalar curvature if the Riemannian metric has constant scalar curvature and the vector field is homothetic. Keywords:Randers metrics, scalar curvature, S-curvatureCategories:53C60, 53B40

7. CMB 2011 (vol 55 pp. 474)

Chen, Bin; Zhao, Lili
 A Note on Randers Metrics of Scalar Flag Curvature Some families of Randers metrics of scalar flag curvature are studied in this paper. Explicit examples that are neither locally projectively flat nor of isotropic $S$-curvature are given. Certain Randers metrics with Einstein $\alpha$ are considered and proved to be complex. Three dimensional Randers manifolds, with $\alpha$ having constant scalar curvature, are studied. Keywords:Randers metrics, scalar flag curvatureCategories:53B40, 53C60

8. CMB 2011 (vol 55 pp. 138)

Li, Benling; Shen, Zhongmin
 Projectively Flat Fourth Root Finsler Metrics In this paper, we study locally projectively flat fourth root Finsler metrics and their generalized metrics. We prove that if they are irreducible, then they must be locally Minkowskian. Keywords:projectively flat, Finsler metric, fourth root Finsler metricCategory:53B40

9. CMB 2009 (vol 52 pp. 132)

Shen, Zhongmin
 On Projectively Flat $(\alpha,\beta)$-metrics The solutions to Hilbert's Fourth Problem in the regular case are projectively flat Finsler metrics. In this paper, we consider the so-called $(\alpha,\beta)$-metrics defined by a Riemannian metric $\alpha$ and a $1$-form $\beta$, and find a necessary and sufficient condition for such metrics to be projectively flat in dimension $n \geq 3$. Categories:53B40, 53C60

10. CMB 2002 (vol 45 pp. 232)

Ji, Min; Shen, Zhongmin
 On Strongly Convex Indicatrices in Minkowski Geometry The geometry of indicatrices is the foundation of Minkowski geometry. A strongly convex indicatrix in a vector space is a strongly convex hypersurface. It admits a Riemannian metric and has a distinguished invariant---(Cartan) torsion. We prove the existence of non-trivial strongly convex indicatrices with vanishing mean torsion and discuss the relationship between the mean torsion and the Riemannian curvature tensor for indicatrices of Randers type. Categories:46B20, 53C21, 53A55, 52A20, 53B40, 53A35
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