Naturally reductive pseudo-Riemannian spaces

A family of naturally reductive pseudo-Riemannian spaces is constructed out of the representations of Lie algebras with ad-invariant metrics. We exhibit peculiar examples, study their geometry and characterize the corresponding naturally reductive homogeneous structure.     

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.     

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.     

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.     

Some Berwald spaces of non-positive flag curvature

In this paper, by using left invariant Riemannian metrics on some three-dimensional Lie groups, we construct some complete non-Riemannian Berwald spaces of non-positive flag curvature and several families of geodesically complete locally Minkowskian spaces of zero constant flag curvature.     

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.     

Abelian gauge theories on homogeneous spaces

An algebraic technique of separation of gauge modes in Abelian gauge theories on homogeneous spaces is proposed. An effective potential for the Maxwell-Chern-Simons theory on S ³ is calculated. A generalization of the Chern-Simons action is suggested and analyzed with the example of SU(3)/U(1) X U(1).     

The existence of homogeneous geodesics in homogeneous pseudo-Riemannian and affine manifolds

It is well known that, in any homogeneous Riemannian manifold, there is at least one homogeneous geodesic through each point. For the pseudo-Riemannian case, even if we assume reductivity, this existence problem is still open. The standard way to deal with homogeneous geodesics in the pseudo-Riemannian case is to use the so-called “Geodesic Lemma”, which is a formula involving the inner product. We shall use a different approach: namely, we imbed the class of all homogeneous pseudo-Riemannian manifolds into the broader class of all homogeneous affine manifolds (possibly with torsion) and we apply a new, purely affine method to the existence problem. In dimension 2, it was solved positively in a previous article by three authors. Our main result says that any homogeneous affine manifold admits at least one homogeneous geodesic through each point. As an immediate corollary, we prove the same result for the subclass of all homogeneous pseudo-Riemannian manifolds.     

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.     

Dolbeault cohomology groups of compact pseudo-Kähler homogeneous manifolds

It is well known that a pseudo-Kähler structure is one of the natural generalizations of a Kähler structure. In this paper, we consider the Dolbeault cohomology groups of compact pseudo-Kähler homogeneous manifolds.     

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.     

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.     

Isotropic submanifolds of pseudo-Riemannian spaces

The family of all the submanifolds of a given Riemannian or pseudo-Riemannian manifold is large enough to classify them into some interesting subfamilies such as minimal (maximal), totally geodesic, Einstein, etc. Most of these have been extensively studied by many authors, but as far as we know, no paper has hitherto been published on the class of isotropic submanifolds. The purpose of this paper is therefore to gain a better understanding of this interesting class of submanifolds that arise naturally in mathematics and physics by studying their relationships with other closely distinguished families.     

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.     

Invariant fourth root Finsler metrics on the Grassmannian manifolds

Fourth root metrics are a special and important class of Finsler metrics, which have been applied to physics. In this paper, we study invariant fourth root Finsler metrics on the Grassmannian manifolds SO(p+q)/SO(p)×SO(q)$SO(p+q)/SO\left(p\right)\times SO\left(q\right)$. By using the results from the theory of invariant polynomials of Lie groups, we obtain a complete classification of such metrics. Further, some invariant 2m$2m$-th root Finsler metrics are also given.