On the connections of sub-Finslerian geometry

A sub-Finslerian manifold is, roughly speaking, a manifold endowed with a Finsler type metric which is defined on a [Formula: see text]-dimensional smooth distribution only, not on the whole tangent manifold. Our purpose is to construct a generalized nonlinear connection for a sub-Finslerian manifold, called [Formula: see text]-connection by the Legendre transformation which characterizes normal extremals of a sub-Finsler structure as geodesics of this connection. We also wish to investigate some of its properties like normal, adapted, partial and metrical.

Vector Bundles and Paracontact Finsler Structures

Almost paracontact and normal almost paracontact Finsler structures on a vector bundle are defined. Finding some conditions, integrability of these structures are studied. Moreover, we define paracontact metric, para- Sasakian and K-paracontact Finsler structures and study some properties of these structures. For a K-paracontact Finsler structure, we find the vertical and horizontal flag curvatures. Then, defining vertical φ-flag curvature, we prove that every locally symmetric para-Sasakian Finsler structure has negative vertical φ-flag curvature. Finally, we define the horizontal and vertical Ricci tensors of a para-Sasakian Finsler structure and study some curvature properties of them.

Finsler-type estimators for the cancer cell population dynamics

We introduce a Finslerian model related to the classical Garner dynamical system, which models the cancer cell population growth. The Finsler structure is determined by the energy of the deformation field-the difference of the fields, which describe the reduced and the proper biological models. It is shown that a certain locally-Minkowski anisotropic Randers structure, obtained by means of statistical fitting, is able to provide a Zermelo-type drift of the overall cancer cell population growth, which occurs due to significant changes within the cancerous process. The geometric background, the applicative advantages and perspective openings of the constructed geometric structure are discussed.

ON FINSLERIZED ABSOLUTE PARALLELISM SPACES

The aim of this paper is to construct and investigate a Finsler structure within the framework of a Generalized Absolute Parallelism (GAP)-space. The Finsler structure is obtained from the vector fields forming the parallelization of the GAP-space. The resulting space, which we refer to as a Finslerized absolute parallelism (parallelizable) space, combines within its geometric structure the simplicity of GAP-geometry and the richness of Finsler geometry, hence is potentially more suitable for applications and especially for describing physical phenomena. A study of the geometry of the two structures and their interrelation is carried out. Five connections are introduced and their torsion and curvature tensors derived. Some special Finslerized parallelizable spaces are singled out. One of the main reasons to introduce this new space is that both absolute parallelism and Finsler geometries have proved effective in the formulation of physical theories, so it is worthy to try to build a more general geometric structure that would share the benefits of both geometries.

Weak Finsler structures and the Funk weak metric

We discuss general notions of metrics and of Finsler structures which we call weak metrics and weak Finsler structures. Any convex domain carries a canonical weak Finsler structure, which we call its tautological weak Finsler structure. We compute distances in the tautological weak Finsler structure of a domain and we show that these are given by the so-called Funk weak metric. We conclude the paper with a discussion of geodesics, of metric balls, of convexity, and of rigidity properties of the Funk weak metric.