We propose a new framework for constructing geometric and physical models on nonholonomic manifolds provided both with Clifford – Lie algebroid symmetry and nonlinear connection structure. Explicit parametrizations of generic off–diagonal metrics and linear and nonlinear connections define different types of Finsler, Lagrange and/or Riemann–Cartan spaces. A generalization to spinor fields and D...

The aim of this paper is to show that the holonomy group of a non-Riemannian Finsler manifold of constant curvature with dimension n > 2 cannot be a compact Lie group and hence it cannot occur as the holonomy group of any Riemannian manifold. This result gives a positive answer to the following problem formulated by S. S. Chern and Z. Shen: Is there a Finsler manifold whose holonomy group is no...

Our main result gives an improved bound on the filling areas of curves in Banach spaces which are not closed geodesics. As applications we show rigidity Pu’s classical systolic inequality and investigate isoperimetric constants normed spaces. The latter has further concerning regularity minimal surfaces Finsler manifolds.

Control Contraction Metrics (CCMs) provide a nonlinear controller design involving an offline search for a Riemannian metric and an online search for a shortest path between the current and desired trajectories. In this paper, we generalize CCMs to Finsler geometry, allowing the use of nonRiemannian metrics. We provide open loop and sampled data controllers. The sampled data control constructio...

In this paper we prove several multiplicity results of t-periodic light rays in conformally stationary spacetimes using the Fermat metric and the extensions of the classical theorems of Gromoll-Meyer and Bangert-Hingston to Finsler manifolds. Moreover, we exhibit some stationary spacetimes with a finite number of t-periodic light rays and compute a lower bound for the period of the light rays w...

A tangent manifold is a pair (M, J) with J a tangent structure (J2 = 0, ker J = im J) on the manifold M . A systematic study of tangent manifolds was done by I. Vaisman in [5]. One denotes by HM any complement of im J := TV . Using the projections h and v on the two terms in the decomposition TM = HM ⊕TV one naturally defines an almost complex structure F on M . Adding to the pair (M, J) a Riem...

Minkowski’s second theorem can be stated as an inequality for n-dimensional flat Finsler tori relating the volume and minimal product of lengths closed geodesics which form a homology basis. In this paper we show how fundamental result promoted to principle holding larger class manifolds. This includes manifolds first Betti number dimension do no necessarily coincide, prime example being case s...

One of the key properties of the length of a curve is its lower semicontinuity : if a sequence of curves γi converges to a curve γ, then length(γ) ≤ lim inf length(γi). Here the weakest type of pointwise convergence suffices. There are higher-dimensional analogs of this semicontinuity for Riemannian (and even Finsler) metrics. For instance, the Besicovitch inequality (see, e.g., [1] and [4]) im...