AUT Journal of Mathematics and Computing
Volume:2 Issue: 2, Oct 2021

Ricci curvature is one of the important geometric quantities in Riemann-Finsler geometry. Together with the S-curvature, one can define a weighted Ricci curvature for a pair of Finsler metric and a volume form on a manifold. One can build up a bridge from Riemannian geometry to Finsler geometry via geodesic fields. Then one can estimate the Laplacian of a distance function and the mean curvature of a metric sphere under a lower weighted Ricci curvature by applying the results in the Riemannian setting. These estimates also give rise to a volume comparison of Bishop-Gromov type for Finsler metric measure manifolds.

In this paper, we study Finsler warped product metric recently introduced by P. Marcal and Z. Shen and find characteristics differential equations for this metric to have vanishing E-curvature. We also prove that if this warped product Finsler metric is projectively flat, then it becomes a Riemannian metric.

In this paper‎, ‎we focus on a class of Finsler metrics which are called general~$(alpha,beta)$-metrics‎: ‎$alpha= sqrt{a_{ij}(x)y^{i}y^{j}}$ is a Riemannian metric and $beta= b_{i}(x)y^{i}$ is a $1$-form. We examine the metrics as weakly isotropic $S$-curvature‎.

The extensions of the Riemannian structure include the Finslerian one, which provided in recent years successful models in various fields like Biology, Physics, GTR, Monolayer Nanotechnology and Geometry of Big Data. The present article provides the necessary notions on tensor spectral data and on the HO-SVD and the Candecomp tensor decompositions, and further study several aspects related to the spectral theory of the main symmetric Finsler tensors, the fundamental and the Cartan tensor. In particular, are addressed two Finsler models used in Langmuir Blodgett Nanotechnology and in Oncology. As well, the HO-SVD and Candecomp decompositions are exemplified for these models and metric extensions of the eigen problem are proposed.

The classical Yamabe problem in Riemannian geometry states that every conformal class contains a metric with constant scalar curvature. In Finsler geometry, the C-convexity is needed in general. In this paper, we study the strong C-convexity of Randers metrics, and provide a result on the Yamabe problem for the metrics of Randers type.

This is a survey on some recent progress in homogeneous Finsler geometry. Three topics are discussed, the classification of positively curved homoge[1]neous Finsler spaces, the geometric and topological properties of homogeneous Finsler spaces satisfying K ≥ 0 and the (FP) condition, and the orbit number of prime closed geodesics in a compact homogeneous Finsler manifold. These topics share the same similarity that the same rank inequality, i.e., rankG ≤ rankH + 1 for G/H with com[1]pact G and H, plays an important role. In this survey, we discuss in each topic how the rank inequality is proved, explain its importance, and summarize some relevant results.

The geometry and analysis on Finsler manifolds is a very important part of Finsler geometry. In this survey article, we introduce some important and fundamental topics in global Finsler geometry and discuss the related properties and the relationships in them. In particular, we optimize and improve the various definitions of Lie derivatives on Finsler manifolds. Further, we also obtain an estimate of lower bound for the non-zero eigenvalues of the Finsler Laplacian under the condition that RicN ≥ K > 0.

The navigation technique is very effective to obtain or classify a Finsler metric from a given a Finsler metric (especially a Riemannian metric) under an action of a vector field on a differential manifold. In this survey, we will survey some recent progress on the navigation problem and conformal vector fields on Finsler manifolds, and their applications in the classifications of some Finsler metrics of scalar (resp. constant) flag curvature.

The notion of generalized Berwald manifolds goes back to V. Wagner [60]. They are Finsler manifolds admitting linear connections on the base manifold such that the parallel transports preserve the Finslerian length of tangent vectors (compatibility condition). Presenting a panoramic view of the general theory we are going to summarize some special problems and results. Spaces of special metrics are of special interest in the generalized Berwald manifold theory. We discuss the case of generalized Berwald Randers metrics, Finsler surfaces and Finsler manifolds of dimension three. To provide the unicity of the compatible linear connection we are looking for, we introduce the notion of the extremal compatible linear connection minimizing the norm of the torsion tensor point by point. The mathematical formulation is given in terms of a conditional extremum problem for checking the existence of compatible linear connections in general. Explicite computations are presented in the special case of generalized Berwald Randers metrics.

This survey is an inspiration of my joint paper with Behzad Najafi published in Science in China. I explain some of interesting results about the unicorn problem.

In this article, we are going to discuss the geometry of the navigation problem on a Finsler manifold. We will give proofs for several important local and global results.

On a real vector space V , a Randers norm Fˆ is defined by Fˆ = ˆα+βˆ, where ˆα is a Euclidean norm and βˆ is a covector. We show that the unit sphere Σ in the Randers space (V, Fˆ) has positive flag curvature, if and only if |βˆ|αˆ < (5 − √ 17)/2 ≈ 0.43845, thus answering a problem proposed by Prof. Zhongmin Shen. Moreover, we prove that the flag curvature of Σ has a universal lower bound −4.