We prove that a Finsler manifold Fm is of constant curvature K = 1 if and only if the unit horizontal Liouville vector field is a Killing vector field on the indicatrix bundle IM of Fm.

In a recent works Liu and Wang 2008; 2007 study the Mannheim partner curves in the three dimensional space. In this paper, we extend the theory of the Mannheim curves to ruled surfaces and define two ruled surfaces which are offset in the sense of Mannheim. It is shown that, every developable ruled surface have a Mannheim offset if and only if an equation should be satisfied between the geodesi...

The starting point of the famous structure theorems on Berwald spaces due to Z.I. Szabó [4] is an observation on the Riemann-metrizability of positive definite Berwald manifolds. It states that there always exists a Riemannian metric on the underlying manifold such that its Levi-Civita connection is just the canonical connection of the Berwald manifold. In this paper we present a new elementary...

The writhe of a space curve fragment is considered for various boundary conditions. An expression for the writhe as a function of arclength for an arbitrary space curve is obtained. The formula is built on the base of closing the tangent indicatrix with a geodesic. The corresponding closure of a curve in 3-space is explicitly constructed. The addition rule for writhe is formulated. A relationsh...

We produce skew loops—loops having no pair of parallel tangent lines—homotopic to any loop in a flat torus or other quotient of Rn . The interesting case here is n = 3 . More subtly for any n , we characterize the homotopy classes that will contain a skew loop having a specified loop τ ⊂ S as tangent indicatrix.

We produce skew loops—loops having no pair of parallel tangent lines—homotopic to any loop in a flat torus or other quotient of Rn . The interesting case here is n = 3 . More subtly for any n , we characterize the homotopy classes that will contain a skew loop having a specified loop τ ⊂ S as tangent indicatrix.

The space of the associative commutative hyper complex numbers, H4, is a 4-dimensional metric Finsler space with the Berwald-Moor metric. It provides the possibility to construct the tensor fields on the base of the analytical functions of the H4 variable and also in case when this analyticity is broken. Here we suggest a way to construct the metric tensor of a 4-dimensional pseudo Riemannian s...