A Finsler metric of constant Gauss curvature $K = 1$ on 2-sphere
We construct a concrete example of constant Gauss curvature $K = 1$ on the 2-sphere having all geodesics closed and of same length.
math.DG↗
arXiv subjects
S. V. Sabau
Publications and source records attributed to S. V. Sabau.
The geometry on the slope of a mountain
The geometry on a slope of a mountain is the geometry of a Finsler metric, called here the {\it slope metric}. We study the existence of globally defined slope metrics on surfaces of revolution as well as the geodesic's behavior. A comparison between Finslerian and Riemannian areas of a bounded region is also studied.
math.DG↗
Surfaces of revolution admitting strongly convex slope metrics
This paper discusses the geometry of a surface endowed with a slope metric. We obtain necessary and sufficient conditions for any surface of revolution to admit a strongly convex slope metric. Such conditions involve certain inequalities for the derivative of the associated function on the Cartesian coordinate and the polar coordinate. In particular, we apply this result to certain well-know surface of revolution.
math.DG↗
Finslerian indicatrices as algebraic curves and surfaces
We show how to construct new Finsler metrics, in two and three dimensions, whose indicatrices are pedal curves or pedal surfaces of some other curves or surfaces. These Finsler metrics are generalizations of the famous slope metric, also called Matsumoto metric.
math.DG↗
The geometry of a positively curved Zoll surface of revolution
In this paper, we study the geometry of the manifolds of geodesics of a Zoll surface of positive Gauss curvature, show how these metrics induce Finsler metrics of constant flag curvature and give some explicit constructions.
math.DG↗
A Gauss-Bonnet Type Formula on Riemann-Finsler surfaces with non-constant indicatrix volume
We prove a Gauss-Bonnet type formula for Riemann-Finsler surfaces of non-constant indicatrix volume and with regular piecewise smooth boundary. We give a Hadamard type theorem for N-parallels of a Landsberg surface.
math.DG↗
On the existence of Generalized Unicorns on Surfaces
This paper addresses the problem of existence of generalized Landsberg structures on surfaces using the Cartan-Kähler Theorem and a Path Geometry approach.
math.DG↗
Moving frames on generalized Finsler structures
We study the relation between an R-Cartan structure α and an (I, J, K)- generalized Finsler structure on a 3-manifold showing the difficulty in finding a general transformation that maps these structures each other. In some particular cases, the mapping can be uniquely determined by geometrical conditions.
math.DG↗
Jacobi stability analysis of dynamical systems -- applications in gravitation and cosmology
The Kosambi-Cartan-Chern (KCC) theory represents a powerful mathematical method for the analysis of dynamical systems. In this approach one describes the evolution of a dynamical system in geometric terms, by considering it as a geodesic in a Finsler space. By associating a non-linear connection and a Berwald type connection to the dynamical system, five geometrical invariants are obtained, with the second invariant giving the Jacobi stability of the system. The Jacobi (in)stability is a natural generalization of the (in)stability of the geodesic flow on a differentiable manifold endowed with a metric (Riemannian or Finslerian) to the non-metric setting. In the present paper we review the basic mathematical formalism of the KCC theory, and present some specific applications of this method in general relativity, cosmology and astrophysics. In particular we investigate the Jacobi stability of the general relativistic static fluid sphere with a linear barotropic equation of state, of the vacuum in the brane world models, of a dynamical dark energy model, and of the Lane-Emden equation, respectively. It is shown that the Jacobi stability analysis offers a powerful and simple method for constraining the physical properties of different systems, described by second order differential equations.
math-ph↗