Landsberg Curvature Research Articles

Finsler warped product metrics with relatively isotropic mean Landsberg curvature

Finsler warped product metrics with relatively isotropic mean Landsberg curvature

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.

Conformal Transformations on General (α,β)-Spaces

In this paper, we study conformal transformations between two almost regular general (α,β)-metrics. By using the method of special coordinate system, the necessary and sufficient conditions for conformal transformations preserving the mean Landsberg curvature are obtained. Further, a rigidity theorem for regular general (α,β)-metrics is proved.

On Almost Rational Finsler Metrics

We study a special class of Finsler metrics which we refer to as Almost Rational Finsler metrics (shortly, AR-Finsler metrics). We give necessary and sufficient conditions for an AR-Finsler manifold (M, F) to be Riemannian. The rationality of some Finsler geometric objects such as Cartan torsion, geodesic spray, Landsberg curvature and S-curvature is investigated. For a particular subfamily of AR-Finsler metrics we have proved that if F has isotropic S-curvature, then the S-curvature vanishes identically; if F has isotropic mean Landsberg curvature, then it is weakly Landsberg; if F is an Einstein metric, then it is Ricci-flat. Moreover, there exists no Randers AR-Finsler metric. Finally, we provide some nontrivial examples of AR-Finsler metrics.

On a class of weakly Landsberg metrics composed by a Riemannian metric and a conformal 1-form

<abstract><p>Without the quadratic restriction, there are many non-Riemannian geometric quantities in Finsler geometry. Among these geometric quantities, Berwald curvature, Landsberg curvature and mean Landsberg curvature are related directly to the famous "unicorn problem" in Finsler geometry. In this paper, Finsler metrics with vanishing weakly Landsberg curvature (i.e., weakly Landsberg metrics) are studied. For the general $ (\alpha, \beta) $-metrics, which are composed by a Riemannian metric $ \alpha $ and a 1-form $ \beta $, we found that if the expression of the metric function doesn't depend on the dimension $ n $, then any weakly Landsberg $ (\alpha, \beta) $-metric with a conformal 1-form must be a Landsberg metric. In the two-dimensional case, the weakly Landsberg case is equivalent to the Landsberg case. Further, we classified two-dimensional Berwald general $ (\alpha, \beta) $-metrics with a conformal 1-form.</p></abstract>

First integrals for Finsler metrics with vanishing $$\chi $$-curvature

We prove that in a Finsler manifold with vanishing \(\chi \)-curvature (in particular with constant flag curvature) some non-Riemannian geometric structures are geodesically invariant and hence they induce a set of non-Riemannian first integrals. Two alternative expressions of these first integrals can be obtained either in terms of the mean Berwald curvature, or as functions of the mean Cartan torsion and the mean Landsberg curvature.

SOME GEOMETRICAL RESULTS ON NEARLY KÄHLER FINSLER MANIFOLDS

This work is intended as an attempt to extend some results of nearly Kählerian Finsler manifolds. We give a condition to generalized $ (a, b, {\bf J})- $manifolds to be weakly Landsberg metric. Furthermore, we find the conditions under which a nearly Kähler Finsler manifold has relatively isotropic Landsberg curvature and relatively isotropic mean Landsberg curvature.

On Homogeneous Randers Metrics

In this paper, we study the curvature features of the class of homogeneous Randers metrics. For these metrics, we first find a reduction criterion to be a Berwald metric based on a mild restriction on their Ricci tensors. Then, we prove that every homogeneous Randers metric with relatively isotropic (or weak) Landsberg curvature must be Riemannian. This provides an extension of well-known Deng-Hu theorem that proves the same result for a homogeneous Berwald-Randers metric of non-zero flag curvature.

Finsler conformal changes preserving the modified Ricci curvature

On a Riemannian manifold, a Liouville transformation is a conformal change which preserves the Ricci tensor. In Finsler geometry, there are various types of Ricci curvature. By adding a Landsberg curvature term to the classical Ricci curvature, we consider the conformal transformations on a Finsler manifold such that this modified Ricci curvature is preserved. We prove that such conformal transformations are homothetic if the space is C-convex. The conformal rigidity for Landsberg surfaces is also obtained.

Properties of Berwald scalar curvature

In this short paper, we prove that a Finsler manifold with vanishing Berwald scalar curvature has zero $\mathbf{E}$-curvature. As a consequence, Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds. This improves a previous result in \cite{Li}. For $(\alpha,\beta)$-metrics on manifold of dimension greater than 2, if the mean Landsberg curvature and the Berwald scalar curvature both vanish, the Berwald curvature also vanishes.

Finsler Warped Product Metrics with Relatively Isotropic Landsberg Curvature

AbstractIn this paper, we study Finsler warped product metrics with relatively isotropic Landsberg curvature. We obtain the differential equations that characterize such metrics. Then we give some examples.

On the Landsberg curvature of a class of Finsler metrics generated from the navigation problem

On the Landsberg curvature of a class of Finsler metrics generated from the navigation problem

S-curvature for a new class of $$(\alpha , \beta )$$ ( α , β ) -metrics

In this paper, we will investigate the S-curvature, the Landsberg curvature, mean Landsberg curvature, Cartan torsion and mean Cartan torsion for recently introduced \((\alpha , \beta )\)-metric \(F=\beta +\frac{a\alpha ^{2}+\beta ^{2}}{\alpha }\) in Piscoran and Mishra (Georgian Math. J. (in press) 2016); where \(\alpha \) is a Riemannian metric; \(\beta \) is an 1-form and \(a\in (0,1]\) is a real positive scalar. We find the necessary and sufficient condition under which this class of Finsler metrics is Riemannian or locally Minkowskian. Finally, we prove that the above mentioned metric has bounded Cartan torsion.

On a class of Finsler metrics with relatively isotropic mean Landsberg curvature

On a class of Finsler metrics with relatively isotropic mean Landsberg curvature

On the class of generalized Landsberg manifolds

In 2000, Bejancu–Farran introduced the class of generalized Landsberg manifolds which contains the class of Landsberg manifolds. In this paper, we prove three global results for generalized Landsberg manifolds. First, we show that every compact generalized Landsberg manifold is a Landsberg manifold. Then we prove that every complete generalized Landsberg manifold with relatively isotropic Landsberg curvature reduces to a Landsberg manifold. Finally, we show that every generalized Landsberg manifold with vanishing Douglas curvature satisfies \(\mathbf{H}=0\).

On Kropina metrics with non-Riemannian curvature properties

In this paper, we first study two significant non-Riemannian quantities Ξ-curvature and H-curvature and show that a Kropina metric is of almost vanishing Ξ-curvature or H-curvature if and only if it is of isotropic S-curvature. Further, we prove that a Kropina metric F is a Douglas metric if and only if the conformally related metric F˜=eκ(x)F is also Douglas metric. Finally, we classify the Kropina metrics with related isotropic weak Landsberg curvature.

Some properties of m-th root Finsler metrics

We prove that every m-th root metric with isotropic mean Berwald curvature reduces to a weakly Berwald metric. Then we show that an m-th root metric with isotropic mean Landsberg curvature is a weakly Landsberg metric. We find necessary and sufficient condition under which conformal β-change of anm-th root metric is locally dually flat. Finally, we prove that the conformal β-change of locally projectively flat m-th root metrics are locally Minkowskian.

Some curvature properties of Cartan spaces with mth root metrics

In this paper, we define some non-Riemannian curvature properties for Cartan spaces. We consider a Cartan space with the mth root metric. We prove that every mth root Cartan space of isotropic Landsberg curvature, or isotropic mean Landsberg curvature, or isotropic mean Berwald curvature reduces to a Landsberg, weakly Landsberg, and weakly Berwald spaces, respectively. Then we show that the mth root Cartan space of almost vanishing H-curvature satisfies H = 0.

On C3-Like Finsler Metrics

In this paper, we study the class of of C3-like Finsler metrics which contains the class of semi-C-reducible Finsler metric. We find a condition on C3-like metrics under which the notions of Landsberg cur- vature and mean Landsberg curvature are equivalent.

ON SEMI-C-REDUCIBILITY OF (α, β)-METRICS

Every non-Riemannian (α, β)-metric on a manifold with dimension n ≥ 3 is semi-C-reducible. Thus the study on semi-C-reducible metrics will enhance our understanding on the physical meaning of (α, β)-metrics. In this paper, we study weakly Landsberg semi-C-reducible metrics with some non-Riemannian curvature properties. Then we find some conditions on semi-C-reducible manifolds under which the notions of isotropic Landsberg curvature and of isotropic mean Landsberg curvature are equivalent.