On Einstein Matsumoto metrics

Einstein metrics are solutions to Einstein field equation in General Relativity containing the Ricci-flat metrics. Einstein Finsler metrics which represent a non-Riemannian stage for the extensions of metric gravity, provide an interesting source of geometric issues and the (α,β)-metric is an important class of Finsler metrics appearing iteratively in physical studies. It is proved that every n-dimensional (n≥3) Einstein Matsumoto metric is a Ricci-flat metric with vanishing S-curvature. The main result can be regarded as a second Schur type Lemma for Matsumoto metrics.     

Affinely equivalent Khler-Finsler metrics on a complex manifold

The purpose of the present paper is to investigate affinely equivalent Khler-Finsler metrics on a complex manifold.We give two facts (1) Projectively equivalent Khler-Finsler metrics must be affinely equivalent;(2) a Khler-Finsler metric is a Khler-Berwald metric if and only if it is affinely equivalent to a Khler metric.Furthermore,we give a formula to describe the affine equivalence of two weakly Khler-Finsler metrics.     

Metrics on complex manifolds

In this note we discuss various canonical metrics on complex manifolds. A generalization of the Bergman metric is proposed and the relations of metrics on moduli spaces are commented. At last, we review some existence theorems of solutions to the Strominger system.     

Balanced Hermitian structures on almost abelian Lie algebras

We study balanced Hermitian structures on almost abelian Lie algebras, i.e. on Lie algebras with a codimension-one abelian ideal. In particular, we classify six-dimensional almost abelian Lie algebras which carry a balanced structure. It has been conjectured in [1] that a compact complex manifold admitting both a balanced metric and an SKT metric necessarily has a Kähler metric: we prove this conjecture for compact almost abelian solvmanifolds with left-invariant complex structures. Moreover, we investigate the behaviour of the flow of balanced metrics introduced in [2] and of the anomaly flow [3] on almost abelian Lie groups. In particular, we show that the anomaly flow preserves the balanced condition and that locally conformally Kähler metrics are fixed points.     

Proper gromov transforms of metrics are metrics

In phylogenetic analysis, a standard problem is to approximate a given metric by an additive metric. Here it is shown that, given a metric D defined on some finite set X and a nonexpansive map f : X → , the one-parameter family of the Gromov transforms D^Δ,f of D relative to f and Δ that starts with D for large values of Δ and ends with an additive metric for Δ = 0 consists exclusively of metrics. It is expected that this result will help to better understand some standard tree reconstruction procedures considered in phylogenetic analysis.     

New results on the geometry of the moduli space of Riemann surfaces

We briefly survey our recent results about the Mumford goodness of several canonical metrics on the moduli spaces of Riemann surfaces,including the Weil-Petersson metric,the Ricci metric, the Perturbed Ricci metric and the Kahler-Einstein metric.We prove the dual Nakano negativity of the Weil-Petersson metric.As applications of these results we deduce certain important results about the L~2-cohomology groups of the logarithmic tangent bundle over the compactifled moduli spaces.     

The equivalence on classical metrics

In this paper we study the complete invariant metrics on Cartan-Hartogs domains which are the special types of Hua domains. Firstly, we introduce a class of new complete invariant metrics on these domains, and prove that these metrics are equivalent to the Bergman metric. Secondly, the Ricci curvatures under these new metrics are bounded from above and below by the negative constants. Thirdly, we estimate the holomorphic sectional curvatures of the new metrics, and prove that the holomorphic sectional curvatures are bounded from above and below by the negative constants. Finally, by using these new metrics and Yau's Schwarz lemma we prove that the new metrics are equivalent to the Einstein-Kahler metric. That means that the Yau's conjecture is true on Cartan-Hartogs domains.     

A new weighted metric: the relative metric I

The M-relative distance, denoted by ρ_M is a generalization of the p-relative distance introduced in [R.-C. Li, SIAM J. Matrix Anal. Appl. 19 (1998) 956-982]. We establish necessary and sufficient conditions under which ρ_M is a metric. In two special cases we derive complete characterizations of this metric. We also present a way of extending the results to metrics sensitive to the domain in which they are defined and find some connections to previously studied metrics. An auxiliary result of independent interest is an inequality related to Pittenger's inequality in Section 4.     

Characterization of the Distance between Subtrees of a Tree by the Associated Tight Span

A characterization is given to the distance between subtrees of a tree defined as the shortest path length between subtrees. This is a generalization of the four-point condition for tree metrics. For this, we use the theory of the tight span and obtain an extension of the famous result by Dress that a metric is a tree metric if and only if its tight span is a tree. Received July 13, 2004     

A computational procedure on higher‐dimensional nilsolitons

In this paper, we develop algorithmic approach to classify nilsoliton metrics on dimension 8. This approach includes finding eigenvalue type of the nilsoliton derivation, the nullity type, the index of the algebra. It can be considered as a continuation of our papers in Abstract and Applied analysis, volume 2013, 1 to 7, (2013), with article ID 871930, and in Journal of Symbolic Computation 50 (2013), 350 ‐ 373. In our previous work, we classified only ordered type, nilsoliton metric Lie algebras ie, the algebras with the derivation type (1 < 2 < 3… < n) in dimension 8 and 9. Here, we consider more general case. We consider such metrics with simple derivations on an indecomposable nilpotent Lie algebra. In one of our previous study, we have already classified nilsoliton metric Lie algebras with nonsingular Gram matrix in dimension 8 in Journal of Symbolic Computation, vol: 50, 350 ‐ 373, 2013. Here, we focus on the metrics with singular Gram matrix. We also develop faster algorithm in classifying such metrics.     

General spherically symmetric Finsler metrics with constant Ricci and flag curvature

In this paper, we investigate the flag curvature of a special class of Finsler metrics called general spherically symmetric Finsler metrics, which are defined by a Euclidean metric and two related 1-forms. We find equations to characterize the class of metrics with constant Ricci curvature (tensor) and constant flag curvature. Moreover, we study general spherically symmetric Finsler metrics with the vanishing non-Riemannian quantity χ-curvature. In particular, we construct some new projectively flat Finsler metrics of constant flag curvature.     

On dually flat square metrics

Square metrics arise from several classification problems in Finsler geometry. They are the rare Finsler metrics to be of excellent geometry properties. It is proved that every non-Riemannian dually flat square metric must be Minkowskian if the dimension ≥3. We also obtain a rigidity result in dually flat Matsumoto metrics.     

Visibility metrics on the infinity boundary of the complex hyperbolic plane

Limiting spherical and horospherical metrics an the infinity boundary of the complex hyperbolic plane are constructed. It is proved that the limiting spherical metric, which automatically is the Carnot–Carathéodory metric, is also a visibility metric, i.e., it belongs to a canonical class of metrics on the infinity boundary. Bibliography: 6 titles.     

S-curvature of isotropic Berwald metrics

Isotropic Berwald metrics are as a generalization of Berwald metrics. Shen proved that every Berwald metric is of vanishing S-curvature. In this paper, we generalize this fact and prove that every isotropic Berwald metric is of isotropic S-curvature. Let F = α + β be a Randers metric of isotropic Berwald curvature. Then it corresponds to a conformal vector field through navigation representation.     

Finsler空间上的Weyl曲率

The Weyl curvature of a Finsler metric is investigated. This curvature constructed from Riemannain curvature. It is an important projective invariant of Finsler metrics. The author gives the necessary conditions on Weyl curvature for a Finsler metric to be Randers metric and presents examples of Randers metrics with non-scalar curvature. A global rigidity theorem for compact Finsler manifolds satisfying such conditions is proved. It is showed that for such a Finsler manifold,if Ricci scalar is negative,then Finsler metric is of Randers type.     

Comparative Properties of Three Metrics in the Space of Compact Convex Sets

Along with the Hausdorff metric, we consider two other metrics on the space of convex sets, namely, the metric induced by the Demyanov difference of convex sets and the Bartels–Pallaschke metric. We describe the hierarchy of these three metrics and of the corresponding norms in the space of differences of sublinear functions. The completeness of corresponding metric spaces is demonstrated. Conditions of differentiability of convex-valued maps of one variable with respect to these metrics are proved for some special cases. Applications to the theory of convex fuzzy sets are given.     

Remarks on two theorems of Qi-Keng Lu

Two alternate arguments in the framework of intrinsic metrics and measures respectively of part of the proof of a famous theorem due to Qi-Keng Lu on Bergman metric with constant negative holomorphic sectional curvature are presented.A relationship between the Lu constant and the holo- morphic sectional curvature of the Bergman metric is given.Some recent progress of the Yau's porblem on the characterization of domain of holomorphy on which the Bergman metric is K(?)hler-Einstein is described.     

An Equivalence of Scalar Curvatures on Hermitian Manifolds

For a Kähler metric, the Riemannian scalar curvature is equal to twice the Chern scalar curvature. The question we address here is whether this equivalence can hold for a non-Kähler Hermitian metric. For such metrics, if they exist, the Chern scalar curvature would have the same geometric meaning as the Riemannian scalar curvature. Recently, Liu–Yang showed that if this equivalence of scalar curvatures holds even in average over a compact Hermitian manifold, then the metric must in fact be Kähler. However, we prove that a certain class of non-compact complex manifolds do admit Hermitian metrics for which this equivalence holds. Subsequently, the question of to what extent the behavior of said metrics can be dictated is addressed and a classification theorem is proved.     

射影Ricci平坦的Kropina度量

本文研究和刻画了射影Ricci平坦的Kropina度量.利用Kropina度量的S-曲率和Ricci曲率的公式,得到了Kropina度量的射影Ricci曲率公式.在此基础上得到了Kropina度量是射影Ricci平坦度量的充分必要条件.进一步,作为自然的应用,本文研究和刻画了由一个黎曼度量和一个具有常数长度的Killing 1-形式定义的射影Ricci平坦的Kropina度量,也刻画了具有迷向S-曲率的射影Ricci平坦的Kropina度量.在这种情形下,Kropina度量是Ricci平坦度量.     

共形平坦的Randers度量

本文研究共形平坦的Randers 度量的性质. 证明了具有数量旗曲率的共形平坦的Randers 度量都是局部射影平坦的, 并且给出了这类度量的分类结果. 本文还证明了不存在非平凡的共形平坦且具有近迷向S 曲率的Randers 度量.