arXiv research

Paper studies curvature in Finsler geometry, proving curvature constancy under isotropy.

We give the explicit formulas of the flag curvatures of left invariant Matsumoto and Kropina metrics of Berwald type. We can see these formulas are different from previous results given recently. Using these formulas, we prove that at any point of an arbitrary connected non-commutative nilpotent Lie group, the flag cur…

Finsleroid-Finsler metrics form an important class of singular (y-local) Finslerian metrics. They were introduced by G. S. Asanov in 2006. As a special case Asanov produced examples of Landsberg spaces of dimension at least three that are not of Berwald type. These are called Unicorns [5]. The existence of regular (y -…

In the present paper we study Randers metics of Berwald type on simply connected 4-dimensional real Lie groups admitting invariant hypercomplex structure. On these spaces, the Randers metrics arising from invariant hyper-Hermitian metrics are considered. Then we give explicit formulas for computing flag curvature of th…

In Theorem 1, we generalize the results of Szabo for Berwald metrics that are not necessary strictly convex: we show that for every Berwald metric F there always exists a Riemannian metric affine equivalent to F. As an application we show (Corollary 3) that every Berwald projectively flat metric is a Minkowski metric; …

In this paper we identify all simply connected 3-dimensional real Lie groups which admit Randers or Matsumoto metrics of Berwald type with a certain underlying left invariant Riemannian metric. Then we give their flag curvatures formulas explicitly.

In this paper, we mainly establish a Cheeger type finiteness theorem for Berwald manifolds. In order to do this, we study the injectivity radius and the convex radius of a Finsler manifold. A Cheeger type estimate on injectivity radii for Finsler manifolds is given and the existence of the center of mass of a Berwald m…

Berwald geometries are Finsler geometries close to (pseudo)-Riemannian geometries. We establish a simple first order partial differential equation as necessary and sufficient condition, which a given Finsler Lagrangian has to satisfy to be of Berwald type. Applied to $(α,β)$-Finsler spaces, respectively $(A,B)$-Finsler…

Recently the present authors introduced a general class of Finsler connections which leads to a smart representation of connection theory in Finsler geometry and yields to a classification of Finsler connections into the three classes. Here the properties of one of these classes namely the Berwald-type connections whic…

We show that, for Finsler spaces with cubic metric, Landsberg spaces are Berwaldian. Also, for decomposable metrics, we determine specific conditions for a space with cubic metric to be of Berwald type, thus refining the result in [6].

Jebsen-Birkhoff theorem extended to Berwald spacetimes.

In this paper, we study generalized Douglas-Weyl $(α, β)$-metrics. Suppose that an regular $(α, β)$-metric $F$ is not of Randers type. We prove that $F$ is a generalized Douglas-Weyl metric with vanishing S-curvature if and only if it is a Berwald metric. Moreover by ignoring the regularity, if $F$ is not a Berwald met…

Study Finsler metrics with vanishing Landsberg curvature.

Defines a new Randers metric based on an existing one.

We study locally conformally Berwald metrics on closed manifolds which are not globally conformally Berwald. We prove that the characterization of such metrics is equivalent to characterizing incomplete, simply-connected, Riemannian manifolds with reducible holonomy group whose quotient by a group of homotheties is clo…

For Finsler spaces (M,F) endowed with m-th root metrics, we provide necessary and sufficient conditions in which they are projectively flat, or projectively related to Berwald/Riemann spaces. We also give a specific characterization for m-th root metrics spaces of Landsberg and of Berwald type.

Study of Randers spacetimes and their Finsler gravity solutions.

We study the existence of a natural `linearisation' process for generalised connections on an affine bundle. It is shown that this leads to an affine generalised connection over a prolonged bundle, which is the analogue of what is called a connection of Berwald type in the standard theory of connections. Various new in…

A Finsler space is called Ricci-quadratic if its Ricci curvature $Ric(x,y)$ is quadratic in $y$. It is called a Berwald space if its Chern connection defines a linear connection directly on the underlying manifold $M$. In this article, we prove that a homogeneous Randers space is Ricci-quadratic if and only if it is of…

Let $G$ be a Lie group equipped with a left invariant Randers metric of Berward type $F$, with underlying left invariant Riemannian metric $g$. Suppose that $\widetilde{F}$ and $\widetilde{g}$ are lifted Randers and Riemannian metrics arising from $F$ and $g$ on the tangent Lie group $TG$ by vertical and complete lifts…

The abstract proves properties of Berwald spaces with non-zero flag curvature.

The paper studies Berwald scalar curvature properties in Finsler geometry.

In this work we study Berwald spacetimes and their vacuum dynamics, where the latter are based on a Finsler generalization of the Einstein's equations derived from an action on the unit tangent bundle. In particular, we consider a specific class of spacetimes which are non-flat generalizations of the very special relat…

The paper classifies Landsberg spherically symmetric Finsler metrics in various dimensions.

New findings show Berwald Finsler spacetimes cannot be metrized.

New metrics defined in Finsler geometry with specific properties.

After summarizing some necessary preliminaries and tools, including Berwald derivative and Lie derivative in pull-back formalism, we present ten equivalent conditions, each of which characterizes Berwald manifolds among Finsler manifolds. These range from Berwald's classical definition to the existence of a torsion-fre…

Finsleroid-Finsler metrics form an important class of singular (y-local) Finsler metrics. They were introduced by G. S. Asanov [2] in 2006. As the special case of the general construction Asanov produced singular (y - local) examples of Landsberg spaces of dimension at least three that are not of Berwald type. The exis…

Characterizes two-dimensional generalized Berwald metrics with vanishing S-curvature.

We formulate the notion of the Finsleroid--Finsler space, including the positive--definite as well as indefinite cases. The associated concepts of angle, scalar product, and the distance function are elucidated. If the Finsleroid--Finsler space is of Landsberg type, then the Finsleroid charge is a constant. The Finsler…

In this paper, we construct a new class of Finsler manifolds called generalized isotropic Berwald manifolds which is an extension of the class of isotropic Berwald manifolds. We prove that every generalized isotropic Berwald manifold is a generalized Douglas-Weyl manifold. On a compact generalized isotropic Berwald man…

Study on Berwald-Weyl curvature with projective invariance and vanishing results.

In this paper, we classify the spherically symmetric Berwald metrics in $\mathbb{R}^n$. For the spherically symmetric Landsberg metrics, we prove that there do not exist any non-Berwald metrics among the regular case. The partial differential equation systems which can respectively characterize the spherically symmetri…

Paper establishes a relation between Berwald scalar curvature and S-curvature.

For a $2$-dimensional non-flat spray we associate a Berwald frame and a $3$-dimensional distribution that we call the Berwald distribution. The Frobenius integrability of the Berwald distribution characterises the Finsler metrizability of the given spray. In the integrable case, the sought after Finsler function is pro…

In this paper the projective curvature invariants of a complex Finsler space are obtained. By means of these invariants the notion of complex Douglas space is then defined. A special approach is devoted to obtain the equivalence conditions that a complex Finsler space should be Douglas. It is shown that any weakly Kähl…

PhD dissertation on Finsler geometry and gravity, focusing on Berwald spaces and exact solutions.

This thesis contains an introduction to the method of average in Finsler geometry. The method is applied to Berwald spaces, obtaining geodesic rigidity conditions. We prove that the Levi-Civita connection of any Riemannian metric affine equivalent to the Berwald metric leaves invariant the indicatrix of th Finsler metr…

In this note it is shown that Berwald spaces admitting the same norm-preserving torsion-free affine connection have the same (weighted) Ricci curvatures. Combing this with Szabó's Berwald metrization theorem one can apply the Cheeger-Gromoll splitting theorem in order to get a full structure theorem for Berwald spaces …

Berwald metrics are particular Finsler metrics which still have linear Berwald connections. Their complete classification is established in an earlier work, [Sz1], of this author. The main tools in these classification are the Simons-Berger holonomy theorem and the Weyl-group theory. It turnes out that any Berwald metr…

Characterizes Kähler-Berwald metrics on complex manifolds.

Locally classifies 4D spherical symmetric Finsler spaces.

We proof that in dimension two, a Finsler metric is Douglas and generalized Berwald, if and only if it is Berwald or a Randers metric $α+ β$, where $β$ is closed and is of constant length with respect to $α$.

We prove that Berwald spaces whose flag curvature is nowhere vanishing are in fact Riemannian spaces. This means that any Berwald space with flag curvature bounded below by a positive number must be also Riemannian. This rigidity result shows the importance of non-Riemannian examples when imposing flag curvature bounds…

We generalize the higher rank rigidity theorem to a class of Finsler spaces, i.e. Berwald spaces. More precisely, we prove that a complete connected Berwald space of finite volume and bounded nonpositive flag curvature with rank at least $2$ whose universal cover is irreducible, is a locally symmetric space or a locall…

The Berwald-Landsberg problem is considered for two dimensional manifolds. We sketch the proof that there are not $\mathcal{C}^5$-regular $y$-global pure Landsberg surfaces. The method used consists on considerer the holonomy representation of the averaged Chern connection and then exhausting all the possible cases, sh…

We show that which that for a Berwald structure, any Riemannian structure that is preserved by the Berwald connection leaves the indicatrix invariant under horizontal parallel transport. We also obtain the converse result: if $({\bf M},F)$ is a Finsler structure such that there exists a Riemannian structure that leaves…

In this paper, we investigate the spherically symmetric Finsler metrics with isotropic S-curvature and obtain a characterized equation. As an application, we prove that these metrics with Douglas type must be Randers metrics or Berwald metrics. This result leads to two classification theorems.