On Ricci tensors of Randers metrics

In this paper, we study Randers metrics and find a condition on the Ricci tensors of these metrics for being Berwaldian. This generalizes Shen’s Theorem which says that every R-flat complete Randers metric is locally Minkowskian. Then we find a necessary and sufficient condition on the Ricci tensors under which a Randers metric of scalar flag curvature is of zero flag curvature.     

Some connectedness problems in positively curved Finsler manifolds

This paper studies some connectedness problems under the positivity hypothesis of various curvatures (k$k$-Ricci and flag curvature). Our approach uses Morse Theory for general end conditions (see [Ioan Radu Peter, The Morse index theorem where the ends are submanifolds in Finsler geometry, Houston J. Math. 32 (4) (2006) 995–1009]). Some previous results related to the flag curvature were obtained in [Ioan Radu Peter, A connectedness principle in positively curved Finsler manifolds, in: H. Shimada, S. Sabau (Eds.), Advanced Studies in Pure Mathematics, Finsler Geometry, Sapporo 2005-In Memory of Makoto Matsumoto, Mathematical Society of Japan, 2007]. Some results from Riemannian geometry are extended to the Finsler category also. The Finsler setting is much more complicated and the difference between Finsler and Riemann settings will be emphasized during the paper.     

Naturally reductive homogeneous Randers spaces

In this paper, we study naturally reductive Randers metrics on homogeneous manifolds. We first prove that naturally reductive Randers metrics are of Berwald type. We then give an explicit formula for the flag curvature of naturally reductive Randers metrics. Finally a necessary and sufficient condition for invariant Randers metrics on homogeneous manifolds being naturally reductive is given.     

Some rigidity theorems for locally symmetrical Finsler manifolds

In this paper we study some rigidity properties for locally symmetrical Finsler manifolds and obtain some results. We obtain the local equivalent characterization for a Finsler manifold to be locally symmetrical and prove that any locally symmetrical Finsler manifold with nonzero flag curvature must be Riemannian. We also generalize a rigidity result due to Akbar-Zadeh.     

On Randers spaces of constant flag curvature

This paper concerns a ubiquitous class of Finsler metrics on smooth manifolds of dimension n. These are the Randers metrics. They figure prominently in both the theory and the applications of Finsler geometry. For n ≥ 3, we consider only those with constant flag curvature. For n = 2, we focus on those whose flag curvature is a (possibly constant) function of position only. We characterize such metrics by three efficient conditions. With the help of examples in 2 and 3 dimensions, we deduce that the Yasuda-Shimada classification of Randers space forms actually addresses only a special case. The corrected classification for that special case is sharp, holds for n ≥ 2, and follows readily from our three necessary and sufficient conditions.     

Homogeneous geodesics in homogeneous Finsler spaces

In this paper, we study homogeneous geodesics in homogeneous Finsler spaces. We first give a simple criterion that characterizes geodesic vectors. We show that the geodesics on a Lie group, relative to a bi-invariant Finsler metric, are the cosets of the one-parameter subgroups. The existence of infinitely many homogeneous geodesics on the compact semi-simple Lie group is established. We introduce the notion of a naturally reductive homogeneous Finsler space. As a special case, we study homogeneous geodesics in homogeneous Randers spaces. Finally, we study some curvature properties of homogeneous geodesics. In particular, we prove that the S-curvature vanishes along the homogeneous geodesics.     

Homogeneous Finsler spaces of negative curvature

We prove that a homogeneous Finsler space with non-positive flag curvature and strictly negative Ricci scalar is a simply connected manifold.     

On the Flag Curvature of Invariant Randers Metrics

In the present paper, the flag curvature of invariant Randers metrics on homogeneous spaces and Lie groups is studied. We first give an explicit formula for the flag curvature of invariant Randers metrics arising from invariant Riemannian metrics on homogeneous spaces and, in special case, Lie groups. We then study Randers metrics of constant positive flag curvature and complete underlying Riemannian metric on Lie groups. Finally we give some properties of those Lie groups which admit a left invariant non-Riemannian Randers metric of Berwald type arising from a left invariant Riemannian metric and a left invariant vector field.      

On weakly symmetric Finsler spaces

In this paper, we study weakly symmetric Finsler spaces. We first study an existence theorem of weakly symmetric Finsler spaces. Then we study some geometric properties of these spaces and prove that any such space can be written as a coset space of a Lie group with an invariant Finsler metric. Finally we show that each weakly symmetric Finsler space is of Berwald type.     

On complex Landsberg and Berwald spaces

In this paper, we study complex Landsberg spaces and some of their important subclasses. The tools of this study are the Chern-Finsler, Berwald, and Rund complex linear connections. We introduce and characterize the class of generalized Berwald and complex Landsberg spaces. The intersection of these spaces gives the so-called G-Landsberg class. This last class contains two other kinds of complex Finsler spaces: strong Landsberg and G-Kähler spaces. We prove that the class of G-Kähler spaces coincides with complex Berwald spaces, in Aikou’s (1996) [1] sense, and it is a subclass of the strong Landsberg spaces. Some special complex Finsler spaces with (α,β)-metrics offer examples of generalized Berwald spaces. Complex Randers spaces with generalized Berwald and weakly Kähler properties are complex Berwald spaces.     

Remarks on Symplectic Connections

This note contains a short survey on some recent work on symplectic connections: properties and models for symplectic connections whose curvature is determined by the Ricci tensor, and a procedure to build examples of Ricci-flat connections. For a more extensive survey, see Bieliavsky et al. [Int. J. Geom. Methods Mod. Phys. 3, 375–420 2006]. This note also includes a moment map for the action of the group of symplectomorphisms on the space of symplectic connections, an algebraic construction of a large class of Ricci-flat symmetric symplectic spaces, and an example of global reduction in a non-symmetric case.     

Uniqueness of maximal surfaces in Generalized Robertson–Walker spacetimes and Calabi–Bernstein type problems

Complete maximal surfaces in Generalized Robertson–Walker spacetimes obeying either the Null Convergence Condition or the Timelike Convergence Condition are studied. Uniqueness theorems that widely extend the classical Calabi–Bernstein theorem, as well as previous results on complete maximal surfaces in Robertson–Walker spacetimes, i.e. the case in which the Gauss curvature of the fiber is a constant, are given. All the entire solutions to the maximal surface differential equation in certain Generalized Robertson–Walker spacetimes are found.     

Extrinsic curvatures of distributions of arbitrary codimension

In this article, using the generalized Newton transformation, we define higher order mean curvatures of distributions of arbitrary codimension and we show that they agree with the ones from Brito and Naveira [F. Brito, A.M. Naveira, Total extrinsic curvature of certain distributions on closed spaces of constant curvature, Ann. Global Anal. Geom., 18 (2000) 371–383]. We also introduce higher order mean curvature vector fields and we compute their divergence for certain distributions and using this we obtain total extrinsic mean curvatures.     

Symmetries of parabolic contact structures

We generalize the concept of locally symmetric spaces to parabolic contact structures. We show that symmetric normal parabolic contact structures are torsion-free and some types of them have to be locally flat. We prove that each symmetry given at a point with non-zero harmonic curvature is involutive. Finally we give restrictions on the number of different symmetries which can exist at such a point.     

On the curvature of warped product Finsler spaces and the Laplacian of the Sasaki–Finsler metrics

As an opening, we prove that a warped product Finsler space F=_F1×fF2$F=F_1{\times }_f{F}_2$ is of constant curvature c$c$ if and only if the base space _F1$F_1$ is also of constant curvature c$c$, the fiber space _F2$F_2$ is of some constant curvature α$\alpha$, and five other partial differential equations are satisfied. A rather similar result is proved for the case of warped product Finsler spaces of scalar curvature. Close relationships between the geometry of the warped product Finsler spaces of constant curvature and the spectral theory of the Laplacian (Laplace–Beltrami operator) of the well-known Sasaki–Finsler metrics of the base space _F1$F_1$ is established by detailed investigation of the above mentioned PDEs. We also define a new tensor for warped product Finsler spaces, which we call a warped-Cartan tensor. Using the tensor we define a new class of warped product Finsler spaces, calling them C-Warped spaces, which contain Landsberg, Berwald, locally Minkowski and Riemannian spaces, but not necessarily all of the constant curvature Finsler spaces of warped product type. Several results are obtained and special cases, for example the case of Riemannian, C-Warped and projectively flat spaces are also considered.     

Unique continuation results for Ricci curvature and applications

Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete conformally compact metrics on such manifolds. Related to this issue, an isometry extension property is proved: continuous groups of isometries at conformal infinity extend into the bulk of any complete conformally compact Einstein metric. Relations of this property with the invariance of the Gauss–Codazzi constraint equations under deformations are also discussed.     

The Ricci flow approach to homogeneous Einstein metrics on flag manifolds

We give a global picture of the normalized Ricci flow on generalized flag manifolds with two or three isotropy summands. The normalized Ricci flow for these spaces reduces to a parameter-dependent system of two or three ordinary differential equations, respectively. Here, we present a qualitative study of these systems’ global phase portrait, which uses techniques of dynamical systems theory. This study allows us to draw conclusions about the existence and the analytical form of invariant Einstein metrics on such manifolds and seems to offer a better insight to the classification problem of invariant Einstein metrics on compact homogeneous spaces.     

Some non-abelian phase spaces in low dimensions

A non-abelian phase space, or a phase space of a Lie algebra, is a generalization of the usual (abelian) phase space of a vector space. It corresponds to a para-Kähler structure in geometry. Its structure can be interpreted in terms of left-symmetric algebras. In particular, a solution of an algebraic equation in a left-symmetric algebra which is an analogue of classical Yang–Baxter equation in a Lie algebra can induce a phase space. In this paper, we find that such phase spaces have a symplectically isomorphic property. We also give all such phase spaces in dimension 4 and some examples in dimension 6. These examples can be a guide for a further development.