Special Finsler Research Articles

Parallel one forms on special Finsler manifolds

<p>In this paper, we investigated the existence of parallel 1-forms on specific Finsler manifolds. We demonstrated that Landsberg manifolds admitting a parallel 1-form had a mean Berwald curvature of rank of at most $ n-2 $. As a result, Landsberg surfaces with parallel 1-forms were necessarily Berwaldian. We further established that the metrizability freedom of the geodesic spray for Landsberg metrics with parallel 1-forms was at least $ 2 $. We figured out that some special Finsler metrics did not admit a parallel 1-form. Specifically, no parallel 1-form was admitted for any Finsler metrics of nonvanishing scalar curvature, among them the projectively flat metrics with nonvanishing scalar curvature. Furthermore, neither the general Berwald's metric nor the non-Riemannian spherically symmetric metrics admited a parallel 1-form. Consequently, we observed that certain $ (\alpha, \beta) $-metrics and generalized $ (\alpha, \beta) $-metrics did not admit parallel 1-forms.</p>

On Minkowskian product of Finsler manifolds

Let (M1,F1) and (M2,F2) be a pair of Finsler manifolds. The Minkowskian product Finsler manifold (M,F) of (M1,F1) and (M2,F2) with respect to a product function f is the product manifold M=M1×M2 endowed with the Finsler metric F=f(K,H), where K=F12,H=F22. In this paper, the Cartan connection and Berwald connection of (M,F) are derived in terms of the corresponding objects of (M1,F1) and (M2,F2). Necessary and sufficient conditions for (M,F) to be Berwald (resp. weakly Berwald, Landsberg, weakly Landsberg) manifold are obtained. Thus an effective method for constructing special Finsler manifolds mentioned above is given.

Tripathi connection in Finsler geometry

Adopting the pullback formalism, a new linear connection in Finsler geometry has been introduced and investigated. Such connection unifies all formerly known Finsler connections and some other connections not introduced so far. Also, our connection is a Finslerian version of the Tripathi connection introduced in Riemannian geometry. The existence and uniqueness of such connection is proved intrinsically. An explicit intrinsic expression relating this connection to Cartan connection is obtained. Some generalized Finsler connections are constructed from Tripathi Finsler connection, by applying the ${P}^{1}$-process and ${C}$-process introduced by Matsumoto. Finally, under certain conditions, many special Finsler connections are given.

A field theory in Randers-Finsler spacetime

Finsler geometry is a natural arena to investigate the physics of spacetimes with local Lorentz violation. The directional dependence of the Finsler metric provides a way to encode the Lorentz-violating effects into the geometric structure of the spacetime. Here, a classical field theory is proposed in a special Finsler geometry, the so-called Randers-Finsler spacetime, where the Lorentz violation is produced by a background vector field. By promoting the Randers-Finsler metric to a differential operator, a Finsler-invariant action for the scalar, gauge and fermions are proposed. The theory contains nonlocal terms, as in the Very Special Relativity based theories. By expanding the Lagrangian, minimal and nonminimal Standard Model Extension terms arise, revealing a perturbative Lorentz violation. For a CPT-even term, the Carrol-Field-Jackiw and derivative extensions are obtained.

English

English

On the extremal compatible linear connection of a Randers space

A linear connection on a Finsler manifold is called compatible to the metric if its parallel transports preserve the Finslerian length of tangent vectors. Generalized Berwald manifolds are Finsler manifolds equipped with a compatible linear connection. Since the compatibility to the Finslerian metric does not imply the unicity of the linear connection in general, the first step of checking the existence of compatible linear connections on a Finsler manifold is to choose the best one to look for. A reasonable choice is introduced in Vincze (J Differ Geom Appl, 2019. arXiv:1909.03096) called the extremal compatible linear connection, which has torsion of minimal norm at each point. Randers metrics are special Finsler metrics that can be written as the sum of a Riemannian metric and a 1-form (they are “translates” of Riemannian metrics). In this paper, we investigate the compatibility equations for a linear connection to a Randers metric. Since a compatible linear connection is uniquely determined by its torsion, we transform the compatibility equations by taking the torsion components as variables. We determine when these equations have solutions, i.e. when the Randers space becomes a generalized Berwald space admitting a compatible linear connection. Describing all of them, we can select the extremal connection with the norm minimizing property. As a consequence, we obtain the characterization theorem in Vincze (Indag Math 26(2):363–379, 2014): a Randers space is a non-Riemannian generalized Berwald space if and only if the norm of the perturbating term with respect to the Riemannian part of the metric is a positive constant.

On conformally doubly warped product Finsler manifold

The aim of the present paper is to introduce the notion of conformally doubly warped product Finsler manifold (CDWPF). For such a Finsler manifold, the coefficients of Barthel connection and its curvature tensor are investigated. The coefficients of Cartan, Berwald, Hashiguchi and Chern (Rund) connections of CDWPF are calculated. Some special Finsler spaces are studied.

Semi-concurrent vector fields in Finsler geometry

In the present paper, we give an answer to a question which is closely related to doubly warped product of Finsler metrics: ‘‘For each n, is there an n-dimensional Finsler manifold (M,F), admitting a non-constant smooth function f on M such that ∂f∂xi∂gij∂yk=0?”. We relate the preceding mentioned condition to different concepts appeared and studied in Finsler geometry. We introduce and investigate the notion of a semi concurrent vector field on a Finsler manifold. We show that some special Finsler manifolds admitting such vector fields turn out to be Riemannian. We prove that Tachibana's characterization of Finsler manifolds admitting a concurrent vector field leads to Riemannian metrics. Various examples for conic Finsler spaces that admit semi-concurrent vector field are presented.

Hypersurface of a Special Finsler Space with Metric

Hypersurface of a Special Finsler Space with Metric

Nonholonomic frames for Finsler spaces with special (alpha, beta)-metric

The purpose of present paper to determine the Finsler spaces due to deformation of special Finsler $ (\alpha, \beta) $ metric. Consequently, we obtain the non-holonomic frame with help of Riemannian metric $ \alpha $ , one form metric $ \beta $ and Randers metric such as forms I. $L(\alpha, \beta)=\alpha \beta + \beta^{2} $ i.e. product of Randers metric and 1-form metric and II. $L(\alpha, \beta)= \alpha^{2} + \alpha \beta$ i.e. product of Randers metric and Riemannian metric.

Nonholonomic frames for Finsler spaces with special (alpha, beta)-metric

The purpose of present paper to determine the Finsler spaces due to deformation of special Finsler $ (\alpha, \beta) $ metric. Consequently, we obtain the non-holonomic frame with help of Riemannian metric $ \alpha $ , one form metric $ \beta $ and Randers metric such as forms I. $L(\alpha, \beta)=\alpha \beta + \beta^{2} $ i.e. product of Randers metric and 1-form metric and II. $L(\alpha, \beta)= \alpha^{2} + \alpha \beta$ i.e. product of Randers metric and Riemannian metric.

Projectively Flat Finsler Space of Douglas Type with Weakly-Berwald (<i>α</i>,<i>β</i>)-Metric

The present article is organized as follows: In the first part, we characterize the important class of special Finsler (α,β)-metric in the form ofL=α+α2/β, whereαis Riemannian metric andβis differential 1-form to be projectively flat. In the second part, we describe condition for a Finsler spaceFnwith an (α,β)-metric is of Douglas type. Further we investigate the necessary and sufficient condition for a Finsler space with an (α,β)-metric to be weakly-Berwald space and Berwald space.

Characterization of Locally Dually Flat Special Finsler Metrics

Characterization of Locally Dually Flat Special Finsler Metrics

Recurrent Finsler manifolds under projective change

The aim of the present paper is to provide an intrinsic investigation of some important special Finsler spaces. These spaces are related to the Berwald [Formula: see text]-curvature tensor of transformed manifold, the Douglas tensor and the projective deviation tensor of projective change. Three spaces are studied and called symmetric, [Formula: see text]-recurrent and [Formula: see text]-recurrent manifolds.

ON THE HYPERSURFACE OF A FINSLER SPACE WITH RANDERS CHANGE OF GENERALIZED (alpha,beta$ METRIC

The study of special Finsler spaces has been introduced by M. Matsumoto (6). The purpose of the present paper is to study hypersurfaces of a Finsler space with (α,β)-metric L � = α + ǫβ + κ β 2 α + β. We have obtained the conditions for this hypersurface to be hyperplane of 1st, 2nd kind but not of 3rd kind.

Average methods and their applications in differential geometry I

In Minkowski geometry the metric features are based on a compact convex body containing the origin in its interior. This body works as a unit ball and its boundary is formed by the unit vectors. Using one-homogeneous extension we have a so-called Minkowski functional to measure the length of vectors. The half of its square is called the energy function. Under some regularity conditions we can introduce an averaged Euclidean inner product by integrating the Hessian matrix of the energy function on the Minkowskian unit sphere. Changing the origin in the interior of the body we have a collection of Minkowskian unit balls together with Minkowski functionals depending on the base points. It is a kind of special Finsler manifolds called a Funk space. Using the previous method we can associate a Riemannian metric as the collection of the averaged Euclidean inner products belonging to different base points. We investigate this procedure in case of Finsler manifolds in general. Central objects of the associated Riemannian structure will be expressed in terms of the canonical data of the Finsler space. We take one more step forward. Randers spaces will be introduced by averaging of the vertical derivatives of the Finslerian fundamental function. The construction will have a crucial role when we apply the general results to Funk spaces together with some contributions to Brickell’s conjecture on Finsler manifolds with vanishing curvature tensor Q .

A Local Classification of Some Special (α,β)-Metrics of Constant Flag Curvature

We classify some special Finsler metrics of constant flag curvature on a manifold of dimension n>2.

Nullity distributions associated to Cartan connection

The Klein-Grifone approach to global Finsler geometry is adopted. The nullity distributions of the three curvature tensors of Cartan connection are investigated. Nullity distributions concerning certain relevant special Finsler spaces are considered. Concrete examples are given whenever the situation needs.

Some theorems in special Finsler spaces and its generalizations in a bivector connected space

Some theorems in special Finsler spaces and its generalizations in a bivector connected space

Energy <i>β</i>-Conformal Change and Special Finsler Spaces

The main aim of the present paper is to establish an intrinsic investigation of the energy β-conformal change of the most important special Finsler spaces, namely, Ch-recurrent, Cv-recurrent, C0-recurrent, Sv-recurrent, quasi-C-reducible, semi-C-reducible, C-reducible, P-reducible, C2-like, S3-like, P2-like and h-isotropic, ···, etc. Necessary and sufficient conditions for such special Finsler manifolds to be invariant under an energy β-conformal change are obtained. It should be pointed out that the present work is formulated in a prospective modern coordinate-free form.