Every Riemannian metric or Finsler on a manifold induces spray via its geodesics. In this paper, we discuss several expressions for the [Formula: see text]-curvature of spray. We show that sprays obtained by projective deformation using S-curvature always have vanishing text]-curvature. Then establish Beltrami Theorem with text].

In this paper we study geometries on the manifold of curves. We define a manifold M where objects c ∈ M are curves, which we parameterize as c : S → lR (n ≥ 2, S is the circle). Given a curve c, we define the tangent space TcM of M at c including in it all deformations h : S → lR of c. We discuss Riemannian and Finsler metrics F (c, h) on this manifold M , and in particular the case of the geom...

In §1 the authors present the basic properties of the scalar product along a curve on a Finsler manifold. In §2 they investigate the variational formulae for the p-energy functional (p ∈ R − {0}). This concept generalises the notions of length (p = 1) and energy (p = 2) of a curve. §3 analyses the extrema of p-energy when the Finsler space has constant curvature. (F3) the fundamental tensor g i...

The present paper deals with an intrinsic investigation of the notion of a concurrent π-vector field on the pullback bundle of a Finsler manifold (M,L). The effect of the existence of a concurrent π-vector field on some important special Finsler spaces is studied. An intrinsic investigation of a particular β-change, namely the energy β-change (L̃(x, y) = L(x, y) + B(x, y)with B := g(ζ, η); ζ bei...

We extend several notions and results from the classical Patterson-Sullivan theory to setting of Anosov subgroups higher rank semisimple Lie groups, working primarily with invariant Finsler metrics on associated symmetric spaces. In particular, we prove equality between Hausdorff dimensions flag limit sets, computed respect a suitable Gromov (pre-)metric manifold, critical exponents subgroups.

It is well known that deformation theorems are the basic tools in critical point theory. They can be derived under a condition of Palais-Smale type ((PS), for short). In the classical setting of a C1 functional f defined on a Banach space X (or a C2 Finsler manifold), we refer to [15]; for a continuous functional f defined on a complete metric space X, we refer to [8], the results of which incl...

This paper is connected with the problem of describing path metric spaces which are homeomorphic to manifolds and biLipschitz homogeneous, i.e., whose biLipschitz homeomorphism group acts transitively. Our main result is the following. LetX = G/H be a homogeneous space of a Lie group G, and let d be a geodesic distance on X inducing the same topology. Suppose there exists a subgroup GS of G whi...