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1. CMB Online first
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 space Categories:53A10, 53B40 |
2. CMB 2012 (vol 57 pp. 209)
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 radius Categories:53B40, 53C22 |
3. CMB 2012 (vol 57 pp. 194)
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 radius Categories:53B40, 53C22 |
4. CMB 2011 (vol 56 pp. 184)
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 quantity Categories:53C60, 53B40 |
5. CMB 2011 (vol 56 pp. 615)
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-curvature Categories:53C60, 53B40 |
6. CMB 2011 (vol 55 pp. 474)
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 curvature Categories:53B40, 53C60 |
7. CMB 2011 (vol 55 pp. 138)
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 metric Category:53B40 |
8. CMB 2009 (vol 52 pp. 132)
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 |
9. CMB 2002 (vol 45 pp. 232)
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 |