arXiv research

Defines new invariants for Riemann-Finsler manifolds, generalizing Preissman's theorem.

Effective field theories with explicit Lorentz violation are intimately linked to Riemann-Finsler geometry. The quadratic single-fermion restriction of the Standard-Model Extension provides a rich source of pseudo-Riemann-Finsler spacetimes and Riemann-Finsler spaces. An example is presented that is constructed from a …

Study geodesics on infinite-dimensional manifolds using Finsler structures.

The correspondence between Riemann-Finsler geometries and effective field theories with spin-independent Lorentz violation is explored. We obtain the general quadratic action for effective scalar field theories in any spacetime dimension with Lorentz-violating operators of arbitrary mass dimension. Classical relativist…

Bipartite Riemann-Finsler geometries with complementary Finsler structures are constructed. Calculable examples are presented based on a bilinear-form coefficient for explicit Lorentz violation.

Study invariants of elliptic curves in LCS manifolds, leading to new phenomena in Riemann-Finsler geometry.

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.

A geometric approach to differential game theory is illustrated. The parallel pursuit is considered as a two-player zero-sum differential game. The optimal strategies of each player is designed based on Riemann-Finsler geometry. Our approach incorporates a closed loop optimal control and the presentation is familiar wi…

The book contains a collection of works on Riemann-Cartan and metric-affine manifolds provided with nonlinear connection structure and on generalized Finsler-Lagrange and Cartan-Hamilton geometries and Clifford structures modelled on such manifolds. The choice of material presented has evolved from various applications…

Solves time-minimizing navigation on a mountain slope using Riemann-Finsler geometry.

We survey the geometry of Lagrange and Finsler spaces and discuss the issues related to the definition of curvature of nonholonomic manifolds enabled with nonlinear connection structure. It is proved that any commutative Riemannian geometry (in general, any Riemann--Cartan space) defined by a generic off--diagonal metr…

Study Riemann-Finsler geometry on tangent bundles of Lie groups with 2D commutator subgroup.

Torsions, curvatures, structure equations and Bianchi identities for locally anisotropic superspaces (containing as particular cases different supersymmetric extensions and prolongations of Riemann, Finsler, Lagrange and Kaluza--Klein spaces) are investigated.

A general approach to formulation of supergravity in higher order anisotropic superspaces (containing as particular cases different supersymmetric extensions and prolongations of Riemann, Finsler, Lagrange and Kaluza--Klein spaces) is given. We analyze three models of locally anisotropic supergravity.

We determine a 2-codimensional CR-structure on the slit tangent bundle $T_0M$ of a Finsler manifold $(M, F)$ by imposing a condition regarding the almost complex structure $Ψ$ associated to $F$ when restricted to the structural distribution of a framed $f$-structure. This condition is satisfied when $(M, F)$ is of scal…

We study the variational problem for $N$-parallel curves on a Finslerian surface by means of Exterior Differential Systems using Griffiths' method. We obtain the conditions when these curves are extremals of a length functional and write the explicit form of Euler-Lagrange equations for this type of variational problem…

In this paper, I will show how to use beta-deformations to deal with dual flatness of Randers metrics. beta-deformations is a new method in Riemann-Finsler geometry, it is introduced by the author(see arxiv:1209.0845). Later on I will provide more applications of the new kind of deformations in Finsler geometry.

In this paper, I will show how to use $β$-deformations to deal with dual flatness of $(α,β)$-metrics. It is a natural continuation of the research on dually flat Randers metrics(see arxiv:1209.1150). $β$-deformations is a new method in Riemann-Finsler geometry, it is introduced by the author(see arxiv:1209.0845).

The aim of this paper is to develop on the 1-jet space J^1(R,M^3) the Finsler-like geometry (in the sense of distinguished (d-) connection, d-torsions and d-curvatures) of the rheonomic Berwald-Moor metric of order three. Some natural geometrical field theories (gravitational and electromagnetic) produced by the preced…

We elaborate an unified geometric approach to classical mechanics, Riemann-Finsler spaces and gravity theories on Lie algebroids provided with nonlinear connection (N-connection) structure. There are investigated the conditions when the fundamental geometric objects like the anchor, metric and linear connection, almost…

The geometric constructions are elaborated on (semi) Riemannian manifolds and vector bundles provided with nonintegrable distributions defining nonlinear connection structures induced canonically by metric tensors. Such spaces are called nonholonomic manifolds and described by two equivalent linear connections also ind…

Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors. By the fundamental result of the theory \cite{V5} such a linear connection must be metrical with respect to the averaged Riemannian metric given by integr…

The study of curvature properties of homogeneous Finsler spaces with $(α, β)$-metrics is one of the central problems in Riemann-Finsler geometry. In the present paper, the existence of invariant vector fields on homogeneous Finsler spaces with square $(α, β)$-metric and Randers changed square $(α, β)$-metric is proved.…

The work extends the A. Connes' noncommutative geometry to spaces with generic local anisotropy. We apply the E. Cartan's anholonomic frame approach to geometrical models and physical theories and develop the nonlinear connection formalism for projective module spaces. Examples of noncommutative generation of anholonom…

In this summary of Habilitation Thesis, it is outlined author's 18 years research activity on mathematical physics, geometric methods in particle physics and gravity, modifications and applications (after defending his PhD thesis in 1994). Ten most relevant publications are structured conventionally into three "strateg…

Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors (compatibi\-li\-ty condition). By the fundamental result of the theory \cite{V5} such a linear connection must be metrical with respect to the averaged Rie…

Study Ricci curvature of homogeneous Finsler spaces with specific metrics.

Methods in Riemann-Finsler geometry are applied to investigate bi-Hamiltonian structures and related mKdV hierarchies of soliton equations derived geometrically from regular Lagrangians and flows of non-stretching curves in tangent bundles. The total space geometry and nonholonomic flows of curves are defined by Lagran…

Extends Tian theorem to Vaisman manifolds for approximations.

In 1972, K. Kenmotsu studied a class of almost contact Riemannian manifolds. Later, such a manifold was called a Kenmotsu manifold. This paper, we studied Kenmotsu manifolds with $(2n+s)$-dimensional $s-$contact metric manifold and this manifold, we have called generalized Kenmotsu manifolds. Necessary and sufficient c…

Study on a new type of manifolds that generalize almost C-manifolds.

Stabilized convex symplectic manifolds are equivalent to flexible Weinstein manifolds.

New class of complex manifolds defined, properties studied.

A selfsimiar manifold is a Riemannian manifold $\left(M,g\right)$ endowed with a homothetic vector field $ξ$. We characterize global selfsimilar manifolds and describe the structure of local selfsimilar manifolds. We prove that any selfsimilar manifold with a potential homothetic vector field is a conical Riemannian ma…

Smoothly embed 3-manifolds in 5-manifolds, simplifying topological to smooth.

The study provides a structure theorem for a new class of noncompact 3-manifolds.

Condition for intersection of real flag manifolds in complex flag manifold.

The paper studies extended quasi-Einstein manifolds with special geometric properties and solitons.

Conditions for flat manifolds as cusp cross-sections in arithmetic hyperbolic manifolds.

Classifies 3-manifolds from simplified (2,0)-trisections of 4-manifolds.

Study on 3D manifolds with specific tensor structures and their properties.

New manifold type PNDP-manifold defined with Einstein warped product structure.

Optimal inequalities for systole, inradius, and volume in hyperbolic 3-manifolds

The paper explores F-manifolds and metrics, constructing canonical structures.

Defines s-manifolds and s-manifolds with corners for symplectic applications.

Embeds 3-manifolds in symplectic 4-manifolds with constraints.

We introduce a new general class of metric f-manifolds which we call (nearly) trans-S-manifolds and includes S- manifolds, C-manifolds, s-th Sasakian manifolds and generalized Kenmotsu manifold studied previously. We prove their main properties and we present many examples which justify their study.

The study provides homological characterizations for $Q$-manifolds and $l_2$-manifolds.