Search

In this paper, we give an estimate of the upper bound of the inner radius for a complete Finsler measure space (M , F , d μ) of weighted Ricci curvature bounded from below, in terms of the d μ -mean curvature bound of the boundary. We also prove the rigidity result that when the boundary is compact, the upper bound is achieved if and only if M is isometric to a forward Finsler geodesic ball of vanishing radical flag curvature. [ABSTRACT FROM AUTHOR]

Recently, Liu-Deng studied projectively flat homogeneous (α , β) -metrics and showed that if these metrics are not Riemannian nor locally Minkowskian, then the Finsler metrics are left invariant Randers metrics on the hyperbolic space H n as a solvable Lie group (Liu and Deng in Forum Math 27:3149–3165, 2015). In this paper, we study homogeneous projectively flat (or projective) general Finsler metrics. First, we prove that homogeneous projectively flat Finsler metrics have vanishing E ¯ -curvature if and only if they have almost isotropic S-curvature if and only if they have relatively isotropic L-curvature. In any cases, the Finsler metric reduces to a locally Minkowskian metric or a Riemannian metric of constant sectional curvature. This yields a classification of homogeneous projective Finsler metrics with the above mentioned non-Riemannian curvatures properties. Finally, we show that Liu-Deng’s Randers metrics are Douglas metrics which have not isotropic S-curvature nor relatively isotropic L-curvature. [ABSTRACT FROM AUTHOR]

In this paper, we study the class of quintic (α; β)-metrics. We show that every weakly Landsberg 5-th root (α; β)-metrics has vanishing S- curvature. Using it, we prove that a quintic (α; β)-metric is a weakly Landsberg metric if and only if it is a Berwald metric. Then, we show that a quintic (α; β)-metric satisβes - = 0 if and only if S = 0. [ABSTRACT FROM AUTHOR]

Finsler manifolds some of whose characteristic tensors are direction independent provide stimulation for current research. In this paper, we show that the direction independence of the mean Landsberg tensor implies the vanishing of these tensor. [ABSTRACT FROM AUTHOR]

The Randers metrics are popular metrics similar to the Riemannian metrics, frequently used in physical and geometric studies. The weak Einstein-Finsler metrics are a natural generalization of the Einstein-Finsler metrics. Our proof shows that if $(M,F)$ is a simply-connected and compact Randers manifold and $F$ is a weak Einstein-Douglas metric, then every special projective vector field is Killing on $(M,F)$. Furthermore, we demonstrate that if a connected and compact manifold $M$ of dimension $n \geq 3$ admits a weak Einstein-Randers metric with Zermelo navigation data $(h,W)$, then either the $S$-curvature of $(M,F)$ vanishes, or $(M,h)$ is isometric to a Euclidean sphere ${\mathbb{S}^n}(\sqrt{k})$, with a radius of $1/\sqrt{k}$, for some positive integer $k$.

In this paper, we investigate the mean Landsberg curvature of two subclasses of (α, β)-metrics. We prove that these subclasses of (α, β)-metrics with vanishing mean Landsberg curvature have vanishing S-curvature. Using it, we prove that these Finsler metrics are weakly Landsbergian if and only if they are Berwaldian. [ABSTRACT FROM AUTHOR]

The characterization of Finsler spaces with Ricci curvature is an ancient and cumbersome one. In this paper, we have derived an expression of Ricci curvature for the homogeneous generalized Matsumoto change. Moreover, we have deduced the expression of Ricci curvature for the aforementioned space with vanishing the S-curvature. These findings contribute significantly to understanding the complex nature of Finsler spaces and their curvature properties. [ABSTRACT FROM AUTHOR]

In this paper, we mainly introduce and study the weakly weighted Einstein–Finsler metrics. First, we show that weakly weighted Einstein–Kropina metrics must be of isotropic S-curvature with respect to the Busemann–Hausdorff volume form under a certain condition about the weight constants. Then we characterize weakly weighted Einstein–Kropina metrics completely via their navigation expressions or via α and β , respectively. In particular, when ν ≠ 0 (or ν = κ = 0 , respectively) and S-curvature with respect to the Busemann–Hausdorff volume form is isotropic, we prove that a Kropina metric determined by navigation data (h, W) is a weakly weighted Einstein metric if and only if the Riemann metric h is a weighted Einstein–Riemann metric. [ABSTRACT FROM AUTHOR]

In this paper, we consider homogeneous Finsler spaces with generalized m-Kropina metric. First, we find the formula for the Ricci curvature of a homogeneous Finsler space with aforesaid metric. Further, we derive the formula of the Ricci curvature for generalized m-Kropina metric having vanishing S-curvature. Finally, we derive the formula for the projective Ricci curvature of generalized m-Kropina Finsler metric. [ABSTRACT FROM AUTHOR]

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

In this paper, we study a class of Finsler measure spaces whose weighted Ricci curvature satisfies {\mathbf {Ric}}_{\infty }=cF^{2}. This class contains all gradient Ricci solitons and Finsler Gaussian shrinking solitons. Thus Finsler measure spaces in this class are called Finsler gradient Ricci solitons. For a Randers measure space, we find sufficient and necessary conditions for this space to be a Finsler gradient Ricci soliton. In particular, we show that Randers-Finsler gradient Ricci solitons must have isotropic S-curvature. Finally, we give an equivalent condition for a Randers measure space to be a Finsler gradient Ricci soliton of constant S-curvature. [ABSTRACT FROM AUTHOR]

In this paper, we give some characterizations on S-curvature in Randers measure spaces, which shows that only the Busemann-Hausdorff volume form can admit a vanishing S-curvature. [ABSTRACT FROM AUTHOR]

In this article, we have focused on the study of S-curvature of Randers-Matsumoto metric on a homogeneous Finsler space. We have deduced the condition for an isometry of Finsler homogeneous space with Randers-Matsumoto metric to be an isometry of Riemannian homogeneous space and proved that the group of isometries of Finsler space are closed subgroups of that of Riemannian space. We have examined the existence of an invariant vector field. Further, we have derived the formula for S-curvature on the reductive homogeneous space, discussed the condition for isotropic S-curvature, and derived the S-curvature of the Randers-Matsumoto metric for the homogenous space by using S-curvature formula. [ABSTRACT FROM AUTHOR]

Background/purpose: Calcium hydroxide [Ca(OH)2], a wildly used intracanal medicament, should be completely removed from the root canal before obturation to avoid negative effects on the treatment. This study evaluated the effectiveness of conventional needle irrigation (CNI), passive ultrasonic activation (PUI), photon-induced photoacoustic streaming (PIPS), XP-endo Finisher (XP) and EDDY for the removal of Ca(OH)2 from S-shaped root canal. Materials and methods: Eighty-four S-shaped root canals in resin blocks were prepared using Protaper Gold up to size #25/0.08 and filled with Ca(OH)2. Five groups were established according to the removal techniques (n = 16): CNI, PUI, PIPS, XP and EDDY group. The positive and negative control group (n = 2) were also established. Sodium hypochlorite 3% was used as the irrigant. Digital radiographs were used to measure the remaining Ca(OH)2. Statistical analysis of the data was performed by using the Kruskal–Wallis test, followed by Dunn's post hoc test with Bonferroni correction (α = 0.05). Results: All the tested techniques completely removed Ca(OH)2 from the straight portion and coronal curve of the S-shaped root canal in 100% of cases. Regarding the apical curve, PUI, EDDY, PIPS and XP removed significantly more Ca(OH)2 than CNI (P 0.05). The complete clearance of Ca(OH)2 from the apical curve was observed in 75%, 62.5%, 56.3%, 43.8% and 0% of cases of PUI, EDDY, PIPS, XP and CNI group, respectively. Conclusion: Irrigant activation enhanced Ca(OH)2 removal from the apical region of the S-shaped root canal. CNI was significantly less effective than all activation techniques.

The characterization of Finsler spaces with Ricci curvature is an ancient and cumbersome one. In this paper, we have derived an expression of Ricci curvature for the homogeneous generalized Matsumoto change. Moreover, we have deduced the expression of Ricci curvature for the aforementioned space with vanishing the S-curvature. These findings contribute significantly to understanding the complex nature of Finsler spaces and their curvature properties.

This paper discusses the sprays of isotropic curvature. We first determine the relationship between χ-curvatures of two projectively related sprays. Based on this, we find an approach to construct sprays of isotropic curvature and find infinitely many sprays of isotropic curvature via some known sprays of isotropic curvature. In particular, by using famous Funk metric Θ, we can construct infinitely many sprays of isotropic curvature, some of which can be induced by Finsler metrics, but others cannot be induced by any Finsler metric. [ABSTRACT FROM AUTHOR]

The class of warped product metrics can often be interpreted as key space models for general theory of relativity and in the theory of space-time structure. In this paper, we study one of the most important non-Riemannian quantities in Finsler geometry which is called the S-curvature. We examined the behavior of the S-curvature in the Finsler warped product metrics. We are going to prove that every Finsler warped product metric R × R n has almost isotropic S -curvature if and only if it is a weakly Berwald metric. Moreover, we show that every Finsler warped product metric has isotropic S -curvature if and only if S -curvature vanishes. [ABSTRACT FROM AUTHOR]

In this paper, we study the class of cubic (α, β)-metrics. We show that every weakly Landsberg cubic (α, β)-metric has vanishing S-curvature. Using it, we prove that cubic (α, β)-metric is a weakly Landsberg metric if and only if it is a Berwald metric. This yields an extension of the Matsumoto’s result for Landsberg cubic Finsler metrics. [ABSTRACT FROM AUTHOR]

In the present paper, we study one of the central problems of Finsler-Geometry, namely the curvature properties of a certain class of homogeneous Finsler spaces. In the first part we describe the existence of some invariant vector fields on a homogeneous Finsler space. In the second part, we derive the formula of S-curvature and prove that homogeneous Finsler space with almost isotropic S-curvature must have vanishing Scurvature. Finally, we deduce the formula of the mean Berwald curvature. [ABSTRACT FROM AUTHOR]

In this paper, we find necessary and sufficient conditions under which a spherically symmetric metric with isotropic S-curvature or isotropic E-curvature is invariant under conformal transformations. As an application, we solve one of the open problems presented by Shen and find a class of metrics of isotropic E-curvature which is not of isotropic S-curvature. [ABSTRACT FROM AUTHOR]

Calcium hydroxide [Ca(OH) 2 ], a wildly used intracanal medicament, should be completely removed from the root canal before obturation to avoid negative effects on the treatment. This study evaluated the effectiveness of conventional needle irrigation (CNI), passive ultrasonic activation (PUI), photon-induced photoacoustic streaming (PIPS), XP-endo Finisher (XP) and EDDY for the removal of Ca(OH) 2 from S-shaped root canal. Eighty-four S-shaped root canals in resin blocks were prepared using Protaper Gold up to size #25/0.08 and filled with Ca(OH) 2. Five groups were established according to the removal techniques (n = 16): CNI, PUI, PIPS, XP and EDDY group. The positive and negative control group (n = 2) were also established. Sodium hypochlorite 3% was used as the irrigant. Digital radiographs were used to measure the remaining Ca(OH) 2. Statistical analysis of the data was performed by using the Kruskal–Wallis test, followed by Dunn's post hoc test with Bonferroni correction (α = 0.05). All the tested techniques completely removed Ca(OH) 2 from the straight portion and coronal curve of the S-shaped root canal in 100% of cases. Regarding the apical curve, PUI, EDDY, PIPS and XP removed significantly more Ca(OH) 2 than CNI (P < 0.05), with no significant differences among these four groups (P > 0.05). The complete clearance of Ca(OH) 2 from the apical curve was observed in 75%, 62.5%, 56.3%, 43.8% and 0% of cases of PUI, EDDY, PIPS, XP and CNI group, respectively. Irrigant activation enhanced Ca(OH) 2 removal from the apical region of the S-shaped root canal. CNI was significantly less effective than all activation techniques. [ABSTRACT FROM AUTHOR]

We endow the disc D = { (x 1 , x 2) ∈ R 2 : x 1 2 + x 2 2 < 4 } with a Poincaré-type Randers metric F λ , λ ∈ [ 0 , 1 ] that 'linearly' interpolates between the usual Riemannian Poincaré disc model ( λ = 0 , having constant sectional curvature - 1 and zero S-curvature) and the Finsler–Poincaré metric ( λ = 1 , having constant flag curvature - 1 / 4 and constant S-curvature with isotropic factor 1/2), respectively. Contrary to our intuition, we show that when λ ↗ 1 , both the flag and normalized S-curvatures of the metric F λ blow up close to ∂ D for some particular choices of the flagpoles. [ABSTRACT FROM AUTHOR]

In this paper we study the Riemannian geometry of simple Lie groups SO(3;R), SL(2;R) and SO(1; 3), equipped with a left invariant Rie-mannian metric. We consider left invariant Randers metrics induced by these left invariant Riemannian metrics. Then, in each case, we obtain the S-curvature and show that although these Randers metrics are not of Berwald or Douglas type but in the case of SO(3;R) it is of almost isotropic S-curvature. Finally, we give the S-curvature of left invariant Randers metrics on four-dimensional Einstein Lie groups. [ABSTRACT FROM AUTHOR]

In this paper, we find a new non-Riemannian quantity for (α, β)-metrics that is closely related to the S-curvature. We call it the Then we show that an (α, β)-metric is Riemannian if and only if S-curvature. Randers metric, we find the relation between S-curvature and S-curvature. [ABSTRACT FROM AUTHOR]

I show that the S-curvature of a Finsler space vanishes if and only if the E-curvature vanishes if and only if the Berwald scalar curvature vanishes; and I extend these results to the case in which these objects are isotropic. [ABSTRACT FROM AUTHOR]

Let F = α 2 β be a Kropina metric on a measure space (M , m) with the navigation data (h , W). In this paper, we give an equivalent characterization for (M , F , m) to be a gradient almost Ricci soliton, which implies that every gradient almost Ricci soliton has vanishing S B H -curvature. Based on this, we prove that (M , F , m) is a gradient almost Ricci soliton if and only if (M , h , f) is a Riemannian gradient almost Ricci soliton, W is a unit Killing vector field on M and f satisfies a differential equation, where f is the potential function f of the measure m (see Theorem 1.2). As an application, we construct some new shrinking, steady and expanding gradient Ricci solitons. [ABSTRACT FROM AUTHOR]

In this paper, we study Finsler metrics with weak, isotropic and flat projective Ricci curvature (briefly, PRic-curvature). First, we prove a rigidity result that shows that for a complete Finsler manifold, inequality PRic Ric holds if and only if S = 0. Then, we show that the Mstsumoto metric is of weak PRic- curvature if and only if it is a PRic-at metric. We characterize projective Ricci flat Matsumoto metrics with constant length one-forms. In this case, we show that the Matsumoto metric reduces to a Ricci flat metric. Finally, we prove that a Matsumoto metric is PRic-reversible if and only if it is PRic-quadratic. [ABSTRACT FROM AUTHOR]

The study of curvature properties of homogeneous Finsler spaces with (α; α)-metrics is one of the central problems in Riemann-Finsler geometry. In this paper, we consider a homogeneous Finsler space with Randers change of square metric. First, we establish an explicit formula for Ricci curvature of a homogeneous Finsler space with the metric under consideration. Next, we find a necessary and sufficient condition under which a homogeneous Finsler space with Randers change of square metric metric is of vanishing S-curvature. A formula for Ricci curvature of a homogeneous Finsler space with aforesaid metric having vanishing S-curvature is established. Finally, we prove that the afore said space having vanishing S-curvature and negative Ricci curvature must be Riemannian. [ABSTRACT FROM AUTHOR]

One of the most important problems in Finsler geometry is to classify Finsler metrics of scalar flag curvature. In this paper, we study the classification problem of Randers metrics of scalar flag curvature. Under the condition that β is a Killing 1-form, we obtain some important necessary conditions for Randers metrics to be of scalar flag curvature. [ABSTRACT FROM AUTHOR]

We prove the existence of an invariant vector field on a homogeneous Finsler space with exponential metric, and we derive an explicit formula for the S-curvature of a homogeneous Finsler space with exponential metric. Using this formula, we obtain a formula for the mean Berwald curvature of such a homogeneous Finsler space. [ABSTRACT FROM AUTHOR]

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 a homogeneous Finsler space with Randers changed square metric has been proved. Further, an explicit formula for S- curvature of Randers changed square metric has been established. Finally, using the formula of S-curvature, the mean Berwald curvature of afore said (α, β)-metric has been calculated. [ABSTRACT FROM AUTHOR]

We obtain complex Hessian and Laplacian comparison theorems on Kähler Finsler manifolds with holomorphic bisectional curvature bounded from below. As applications, we prove Bishop type volume comparison theorem and Myers type theorem for Kähler Finsler manifolds. Moreover, we also obtain some comparison theorems for the first eigenvalue of Finsler–Laplacian and give a sharp upper estimate of the bottom of the spectrum for Kähler Finsler manifolds whose holomorphic bisectional curvature is bounded from below by - 1 . [ABSTRACT FROM AUTHOR]

The study of curvature properties of homogeneous Finsler spaces with (α , β) -metrics is one of the central problems in Riemann–Finsler geometry. In this paper, the existence of invariant vector fields on a homogeneous Finsler space with square metric is proved. Further, an explicit formula for S -curvature of a homogeneous Finsler space with square metric is established. Finally, using the formula of S -curvature, the mean Berwald curvature of aforesaid (α , β) -metric is calculated. [ABSTRACT FROM AUTHOR]

projective Ricci curvature. Then we characterize isotropic projective Ricci curvature of Randers metrics. we also show that Randers metrics are PRic-reversible if and only if they are PRic-quadratic../files/site1/files/0Abstract2.pdf

In the present paper, we prove that a general (α , β) -metric F is of isotropic S -curvature if and only if it is of isotropic E -curvature under an extra condition on α and β. [ABSTRACT FROM AUTHOR]

In this paper, we study homogeneous Einstein Kropina metric. First, we characterize the sufficient and necessary condition for a homogeneous Kropina metric to be Einstein and with vanishing S-curvature.Further, we study the conformal deformation of the metric F(α,β) = K(α,β) + ε(x)β, where K(α,β) = α2/β and ε(x) depends on the position only. Finally, we prove that the conformal deformation of Kropina metric is single colored. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new class of a family of... Finsler metrics, defined as F = ..., where ... is a Riemannian metric; ... are constants. We study some important properties for this family of (α; β)-metric, such as: the projectively atness for the Finsler space endowed with a type of this metric, the nonholonomic frame for the Finsler space endowed with this family of (α; β)-metrics; the S-curvature, the Landsberg curvature and also the bounded Cartan torsion. [ABSTRACT FROM AUTHOR]

In this paper, first we prove the existence of invariant vector field on a homogeneous Finsler space with infinite series (α, β)-metric. Next, we deduce an explicit formula for the the S-curvature of homogeneous Finsler space with infinite series (α, β)-metric. Using this formula, we further derive the formula for mean Berwald curvature of the homogeneous Finsler space with this metric. [ABSTRACT FROM AUTHOR]

In this paper, we derive the formula for S-curvature of homogeneous Finsler space with special (α,β)-metric F = α+ β²/β and further proved that the almost isotropic S-curvature must vanishes S-curvature. Also we find the explicit formula of the mean Berwald curvature. [ABSTRACT FROM AUTHOR]

In this short paper, we study a symmetric covariant tensor in Finsler geometry, which is called the mean Berwald curvature. We first investigate the geometry of the fibres as the submanifolds of the tangent sphere bundle on a Finsler manifold. Then we prove that if the mean Berwald curvature is isotropic along fibres, then the Berwald scalar curvature is constant along fibres. [ABSTRACT FROM AUTHOR]

An ( α , β ) -metric is defined by a Riemannian metric α and 1-form β . We have characterized a class of two-dimensional ( α , β ) -metrics of isotropic (thus vanishing) S-curvature. In this paper, we determine the local structure of those metrics and show that those metrics are Einsteinian (equivalently, of isotropic flag curvature) but not Ricci-flat in general. [ABSTRACT FROM AUTHOR]

In this paper, we study S-closed conformal transformations, i.e., conformal transformations preserving the S-curvature invariant. Using the maximum principle, we prove that such transformation must be a homothety unless the Finsler manifold is Riemannian. [ABSTRACT FROM AUTHOR]

In this paper we obtain some Cheeger-Gromov-Taylor type compactness theorems for a forward complete and connected Finsler manifold of dimensional n ≥ 2 via weighted Ricci curvatures. The proofs are based on the index form of a minimal unit speed geodesic segment, Bochner-Weitzenböck formula and Hessian comparison theorem. [ABSTRACT FROM AUTHOR]

For an ( α, β)-metric (non-Randers type) of isotropic S-curvature on an n-dimensional manifold with non-constant norm ║ β║, we first show that n = 2, and then we characterize such a class of two-dimensional ( α, β)-manifolds with some PDEs, and also construct some examples for such a class. [ABSTRACT FROM AUTHOR]

Let ( M , F , d μ ) be a Finsler manifold with the Ricci curvature bounded below by a positive number and constant S-curvature. We prove that, if the first eigenvalue of the Finsler⁻Laplacian attains its lower bound, then M is isometric to a Finsler sphere. Moreover, we establish a comparison result on the Hessian trace of the distance function.