Flag Curvature Research Articles

The isoperimetric problem in Randers Poincaré disc

It is known that a simply connected Riemann surface satisfies the isoperimetric equality if and only if it has constant Gaussian curvature. In this paper, we show that the circles centered at origin in the Randers Poincaré disc satisfy the isoperimetric equality with respect to different volume forms however, these Randers metrics do not necessarily have constant (negative) flag curvature. In particular, we show that Osserman’s result [12] of the Riemannian case cannot be extended to the Finsler geometry as such.

On Finsler Surfaces with Isotropic Main Scalar

Let (M,F) be a Finsler surface with the isotropic main scalar I=I(x). The well-known Berwald’s theorem states that F is a Berwald metric if and only if it has a constant main scalar I=constant. This ensures a kind of equality of two non-Riemannian quantities for Finsler surfaces. In this paper, we consider a positively curved Finsler surface and show that H=0 if and only if I=0. This provides an extension of Berwald’s theorem. It follows that F has an isotropic scalar flag curvature if and only if it is Riemannian. Our results yield an infrastructural development of some equalities for two-dimensional Finsler manifolds.

Some examples of Finsler metrics with special curvature properties

In this paper, we give a lot of new non-projectively flat Finsler warped product metrics with zero flag curvature. We also manufacture new generalized Douglas–Weyl (GDW) metrics which are nontrivial in the sense that these metrics are neither Douglas metrics nor Weyl metrics via hyperbolic spaces.

ON HOMOGENEOUS 2-DIMENSIONAL FINSLER MANIFOLDS WITH ISOTROPIC FLAG CURVATURES

We show that every Finsler surface with isotropic main scalar and isotropic flag curvature is Riemannian or relatively constant Landsberg metric. Using it, we prove that every homogeneous Finsler surface with isotropic flag curvature and isotropic main scalar is Riemannian or locally Minkowskian.

Black hole solutions with constant Ricci scalar in a model of Finsler gravity

Ricci scalar being zero is equivalent to the vacuum field equation in Finsler space-time. The Schwarzschild metric can be concluded from the field equation's solution if the space-time conserves spherical symmetry. This research aims to investigate Finslerian Schwarzschild-de Sitter space-time. Recent studies based on Finslerian space-time geometric models are becoming more prevalent because the local anisotropic structure of space-time influences the gravitational field and gives rise to modified cosmological relations. We suggest a gravitational field equation with a non-zero cosmological constant in Finslerian geometry and apprehend that the presented Finslerian gravitational field equation corresponds to the non-zero Ricci scalar. In Finsler geometry, the peer of spherical symmetry is the Finslerian sphere. Assuming space-time to conserve the “Finslerian sphere” symmetry, the counterpart of the Riemannian sphere (Finslerian sphere) must have a constant flag curvature (λ). It is demonstrated that the Finslerian covariant derivative of the geometric part of the gravitational field equation is preserved under a condition using the Chern connection. According to the string theory, string clouds can be defined as a pool of strings made due to symmetry breaking in the universe's early stages. We find that for λ ≠ 1, this solution resembles a black hole surrounded by a cloud of strings. Furthermore, we investigate null and time-like geodesics for λ = 1. In this regard, the photon geodesics are obtained that are the closest paths to the photon sphere of the first photons visible at the black hole shadow limit. Also, circular orbit conditions are obtained for the effective potential.

Conformally related invariant (α, β)-metrics on homogeneous spaces

In this paper, we give the flag curvature formula of general (α, β)-metrics of Berwald type. We study conformally related (α, β)-metrics, especially general (α, β)-metrics that are conformally related to invariant (α, β)-metrics. Also, a necessary and sufficient condition for a Finsler metric conformally related to an (α, β)-metric is given and conformally related Douglas Randers metrics are studied. Finally, we present some examples of conformally related (α, β)-metrics.

Pseudo-Finsler Radially Symmetric Spaces

We introduce the concept of radially symmetric pseudo-Finsler spaces, which generalize the notion of symmetric Finsler spaces, and prove that this concept is equivalent to the preservation of flag curvature by parallel transport together with reversibility. As a consequence, reversible pseudo-Finsler manifolds with constant flag curvature are radially symmetric.

Some curvature properties of spherically symmetric Finsler metrics

In this paper, we study some important Remannian and non-Riemannian curvature properties of spherically symmetric Finsler metrics. Under a condition on the geodesic coefficient, we find the necessary and sufficient conditions under which spherically symmetric metrics are of scalar flag curvature, [Formula: see text]-quadratic or projectively Ricci-flat. For spherically symmetric metrics of relatively isotropic Landsberg curvature, we find the necessary and sufficient conditions under which these metrics are of constant flag curvature or Ricci-quadratic. Finally, we prove a rigidity result that every spherically symmetric metric of relatively isotropic Landsberg curvature is Ricci-quadratic if and only if it is a Berwald metric. Moreover, if the Finsler metric is negatively complete then it reduces to a Riemannian metric.

Traveling along horizontal broken geodesics of a homogeneous Finsler submersion

In this paper, we discuss how to travel along horizontal broken geodesics of a homogeneous Finsler submersion, i.e., we study, what in Riemannian geometry was called by Wilking, the dual leaves. More precisely, we investigate the attainable sets Aq(C) of the set of analytic vector fields C determined by the family of horizontal unit geodesic vector fields C to the fibers F={ρ−1(c)} of a homogeneous analytic Finsler submersion ρ:M→B. Since reverse of geodesics don't need to be geodesics in Finsler geometry, one can have examples on non compact Finsler manifolds M where the attainable sets (the dual leaves) are no longer orbits or even submanifolds. Nevertheless we prove that, when M is compact and the orbits of C are embedded, then the attainable sets coincide with the orbits. Furthermore, if the flag curvature is positive then M coincides with the attainable set of each point. In other words, fixed two points of M, one can travel from one point to the other along horizontal broken geodesics.In addition, we show that each orbit O(q) of C associated to a singular Finsler foliation coincides with M, when the flag curvature is positive, i.e., we prove Wilking's result in Finsler context. In particular we review Wilking's transversal Jacobi fields in Finsler case.

The geometry of geodesic invariant functions and applications to Landsberg surfaces

<p>In this paper, for a given spray $ S $ on an $ n $-dimensional manifold $ M $, we investigated the geometry of $ S $-invariant functions. For an $ S $-invariant function $ {\mathcal P} $, we associated a vertical subdistribution $ {{\mathcal V}}_{\mathcal P} $ and found the relation between the holonomy distribution and $ {{\mathcal V}}_{\mathcal P} $ by showing that the vertical part of the holonomy distribution is the intersection of all spaces $ {{\mathcal V}}_{ {\mathcal F}_S} $ associated with $ {\mathcal F}_S $ where $ {\mathcal F}_S $ is the set of all Finsler functions that have the geodesic spray $ S $. As an application, we studied the Landsberg Finsler surfaces. We proved that a Landsberg surface with $ S $-invariant flag curvature is Riemannian or has a vanishing flag curvature. We showed that for Landsberg surfaces with non-vanishing flag curvature, the flag curvature is $ S $-invariant if and only if it is constant; in this case, the surface is Riemannian. Finally, for a Berwald surface, we proved that the flag curvature is $ H $-invariant if and only if it is constant.</p>

On harmonic and asymptotically harmonic Finsler manifolds

In the present paper we introduce and investigate some types of harmonic Finsler manifolds and find out the interrelation between them. We give some characterizations of such spaces in terms of the Finsler mean curvature of geodesic spheres and the Finsler Laplacian of the distance function induced by the Finsler metric. We investigate some properties of the Finsler mean curvature of geodesic spheres of different radii. We prove that certain harmonic Finsler manifolds are of isotropic S-curvature and isotropic Ricci curvature. Moreover, we give some examples of non-Riemannian Finsler harmonic manifolds of constant flag curvature and constant S-curvature. Finally, we provide a technique to construct harmonic Finsler manifolds of Randers type.

On the non-Riemannian quantity Ξ-curvature of (α,β)-metrics

The non-Riemannian quantity Ξ-curvature which is very useful to characterize the Finsler metrics with constant flag curvature forms an important class in Finsler geometry. In this paper, we classify the (α,β)-metrics with almost isotropic Ξ-curvature.

FINSLER METRICS WITH BUSEMANN CURVATURE BOUNDS

We prove that a Finsler metric has Busemann curvature bounded above (below, respectively) by $\kappa$ if and only if it is the Berwald metric with flag curvature bounded above (below, respectively) by $\kappa$. Combining this with Szabó’s Berwald metrization theorem, we can obtain that such a Finsler metric is affinely equivalent to a Riemannian metric with sectional curvature bounded above (below, respectively) by $\kappa$.

On recurrent Riemannian and Ricci curvatures of Finsler metrics

In this paper, we study a property of Riemannian and Ricci curvatures under which it reproduces itself, namely, recurrent Finsler metrics. We prove that if (M,F) is a recurrent Finsler manifold of non-zero isotropic flag curvature, then F is a Landsberg metric. It follows that Every positively complete 2-dimensional Randers metric is recurrent if and only if it is a Riemannian or locally Minkowskian metric. Next, we study two positive (or negative) projectively related Ricci parallel Finsler metrics on a compact manifold. We show that the projective equivalence is trivial and then the Riemannian curvatures are equal. In the same vein, we explore the class of Ricci-recurrent Randers metrics with closed and conformal form, and show that the related Riemannian metric is Ricci-recurrent if and only if the Randers metric is a Berwald metric. Finally, we find the necessary and sufficient condition under which a Kropina metric be Ricci-recurrent, provided that its one-form is closed and conformal.

ON THE PROJECTIVE ALGEBRA OF MATSUMOTO SPACE

In the present paper we are study of Matsumoto space on the projective algebra and Lie Algebra of the projective group. The projective Algebra of Matsumoto space is characterized as certain Lie sub algebra of the projective algebra. Further, which is devoted to studying the condition of Finsler space of constant flag curvature and vanishing S curvature admits a non Riemannian space of affine projective vector field with Matsumoto metric is Berwald space.

Reversible Finsler metrics of constant flag curvature

Without the restriction of quadratic form as a Riemannian metric, a Finsler metric F=F(x,y) on a smooth manifold M can be reversible (symmetric in y) or not. Reversible Finsler metrics have different properties from Riemannian metrics though it seems they are very close to Riemannian metrics. Hilbert metric is the famous reversible Finsler metric of negative constant flag curvature in the history, and it is projectively flat. Then it is natural to ask the question how to classify reversible projectively flat Finsler metrics of constant flag curvature and give more new examples? In this paper, we answer the above question by giving the classification when the flag curvature K=−1,0,1 respectively. Especially, for the case when K=−1, we show that the only reversible projectively flat Finsler metrics are just Hilbert metrics. For the case when K=1, we give an algebraic way to construct explicit metric function by solving algebraic equations, such as by solving a quartic equation. When the flag curvature is zero, it is much easier to construct reversible projectively flat Finsler metrics than before.

On isotropic Berwald scalar curvature

In this paper, we establish a closer relation between the Berwald scalar curvature and the [Formula: see text]-curvature. In fact, we prove that a Finsler metric has isotropic Berwald scalar curvature if and only if it has weakly isotropic [Formula: see text]-curvature. For Finsler metrics of scalar flag curvature and of weakly isotropic [Formula: see text]-curvature, they have almost isotropic [Formula: see text]-curvature if and only if the flag curvature is weakly isotropic.

Volume comparison theorems in Finsler spacetimes

In a [Formula: see text]-dimensional Lorentz–Finsler manifold with [Formula: see text]-Bakry–Émery Ricci curvature bounded from below, where [Formula: see text], using the Riccati equation techniques, we establish the Bishop–Gromov volume comparison theorem for the so-called standard sets for comparisons in Lorentzian volumes (SCLVs). We also establish the Günther volume comparison theorem for SCLVs when the flag curvature is bounded above.

ON A CLASS OF WEAKLY BERWALD SPECIAL (Α, Β)-METRICS OF SCALAR FLAG CURVATURE

In this paper, we study the Finsler space with special (α, β)-metric is scalar flag curvature and we proved that, if it is weakly Berwald if and only if it is Berwald and vanishes flag curvature. Further, we found that this metric is locally Minkowskian.

On Sprays of Scalar Curvature and Metrizability

Every Finsler metric naturally induces a spray but not so for the converse. The notion for sprays of scalar (resp. isotropic) curvature has been known as a generalization for Finsler metrics of scalar (resp. isotropic) flag curvature. In this paper, a new notion, sprays of constant curvature, is introduced and especially it shows that a spray of isotropic curvature is not necessarily of constant curvature even in dimension \(n\ge 3\). Further, complete conditions are given for sprays of isotropic (resp. constant) curvature to be Finsler metrizable. Based on this result, the local structure is determined for locally projectively flat Berwald sprays of isotropic (resp. constant) curvature which are Finsler metrizable, and some more sprays of isotropic curvature are discussed for their metrizability. Besides, the metrizability problem is also investigated for sprays of scalar curvature under certain curvature conditions.