Connections and Finsler geometry of the structure group of a JB-algebra

One of the key properties of the length of a curve is its lower semi-continuity: if a sequence of curves γi converges to a curve γ, then length(γ) ≤ lim inf length(γi). Here the weakest type of point-wise convergence suffices. There are higher-dimensional analogs of this semi-continuity for Riemannian (and even Finsler) metrics. For instance, the Besicovitch inequality (see, for instance, [1] and [4]) implies that if a sequence of Riemannian metrics di on a manifold M uniformly converges to a Riemannian metric d, then V ol(M,d) ≤ lim inf V ol(M,di). Furthermore, the same is true if the limit metric is Finsler (where one can use any “reasonable” notion of volume for Finsler manifolds); the proof, though, is more involved (see [2], [7]). However, we will give an example of an increasing sequence of Riemannian metrics di on a 2-dimensional disc D, which uniformly converge to a length metric d on D such that Area(D, di) 1 (where by Area(D, d) we mean the 2-dimensional Hausdorff measure). Furthermore, metrics di and d can be realized by a uniformly converging sequence of embeddings of D into R. Our motivation for studying the semi-continuity of the surface area functional came from [3], where a more sophisticated Besicovitch-type inequality for Finsler metrics is shown. The proof is essentially Finsler, even though the inequality makes sense for general length spaces. The counter example undermines a natural approach to proving length-area inequalities for length spaces by means of approximations by Riemannian (more generally, Finsler) metrics. Similar considerations lead to the following question: can every intrinsic metric on a disc be approximated by an increasing sequence of Finsler metrics? There is some evidence suggesting that the answer is likely affirmative in dimension two. However, we will give an example of an intrinsic metric on a 3-dimensional ball such that no neighborhood of the origin admits a Lipschitz bijection to a Euclidean region. In this elementary exposition we present both counter-examples. Unfortunately, people often choose not to publish the results of research that led to counterexamples rather than proofs of desired theorems; as such, even published counterexamples tend to be forgotten. Hence we cannot be confident in complete novelty of the results. At the very least, we use this paper to raise open problems and embed these problems into a new context. The paper is organized as follows. In the rest of the Introduction we give rigorous formulations of the results and outline the proofs. Sections 2 and 3 contain proofs

<p>The present communication has mainly been divided into four sections of which the first section is introductory, the second section deals with R - recurrent of order one. In this section we have derived results telling as to when a - recurrent of order one will be R - recurrent of order one, - recurrent of order one will be a - recurrent of order one. In this section we have also derived the Bianchi’s identity and few more identities which hold in a R - recurrent of order one. The third section deals with R - recurrent of order two. In this section we have observed that the recurrence tensor field is non-symmetric, few more relations and the Bianchi’s identity have been derived in a R - recurrent of order two. In the fourth and the last section we have derived the conditions under which a Landsberg space in a - Finsler space, a - Finsler space is semi - P2- like, a - Finsler space is a - Finsler space, a – Finsler space is P- symmetric, a - Finsler space is P2 like</p>

symmetric Finsler manifold is the generalization of 3-symmetric Riemannian manifold. In this paper we give the definition of 3-symmetric Finsler manifold and characterize 3- symmetric Finsler manifold as homogeneous space. We also give the conditions for the existence of 3-symmetric Finsler metrics on homogeneous spaces and the relation between 3-symmetric Finsler manifold and 3-symmetric Riemannian manifold. In the end, we give the flag curvature and curvature tensor of naturally reductive 3-symmetric Finsler manifold. Keywords: 3-symmetric Finsler manifold; local cubic diffeomorphism; flag curvature MR(2010) Subject Classification: 53C30; 53C60 / CLC number: O186.14 Document code: A Article ID: 1000-0917(2015)04-0607-07 3-symmetric Finsler manifold is the generalization of 3-symmetric Riemannian manifold. It has close relation to complex Finsler manifold (6) . In 1972, Gray gave the definition of pseudo-Riemannian 3-symmetric space and the classification of pseudo-Riemannian 3-symmetric spaces (9) . The theory of pseudo-Riemannian 3-symmetric spaces parallels that of ordinary sym- metric spaces (7,10) to a great extent. However, there are important exceptions. For example, 3-symmetric spaces are automatically almost complex manifolds. In this paper we generalize 3-symmetric Riemannian manifolds to 3-symmetric Finsler man- ifolds. This paper is organized as follows. In Section 1 we give some basic definitions of Finsler manifold. In Section 2 we give the definition of 3-symmetric Finsler manifold. In Section 3 we give the algebraic structure of 3-symmetric Finsler manifold and the condition for the exis- tence of invariant Finsler metric on a homogeneous space which makes the homogeneous space a 3-symmetric Finsler manifold. In Section 4 we give the curvature tensor and flag curvature of naturally reductive 3-symmetric Finsler manifold. In Section 5 we give some examples of 3-symmetric Finsler manifolds. 1 Preliminaries

By associating to a strongly pseudoconvex complex Finsler metric F a Hermitian tensor \({T}\) of type (1, 1), we prove that a weakly Kahler Finsler metric is a Kahler Finsler metric if and only if \({T \equiv 0}\), and a weakly complex Berwald metric is a weakly Kahler Finsler metric if and only if it is a Kahler Finsler metric. We introduce a class of explicitly constructed smooth complex Finsler metrics, called the general complex (\({\alpha, \beta)}\) metric, where \({\alpha}\) is a Hermitian metric and \({\beta}\) is a complex differential form of type (1, 0) on a complex manifold. The holomorphic curvature of the general complex \({(\alpha, \beta)}\) metrics is derived, and a necessary and sufficient condition for these metrics to be weakly Kahler Finsler metrics is obtained. Under the assumption that \({\beta}\) is holomorphic or closed, necessary and sufficient conditions for this class of metrics to be weakly complex Berwald metrics, complex Berwald metrics and Kahler Finsler metrics are obtained, respectively.

On dually flat Finsler metrics

On Asanov's Finsleroid–Finsler metrics as the solutions of a conformal rigidity problem

In the present work, we wanted to find the possible way in order to make equivalence between non-commutative and Finsler geometries as two useful mathematical tools. Based on this purpose, we were concerned to search this possibility by investigating the massive gravity black holes. Firstly the Lagrangian of system is introduced and then it is rewritten in the non-commutative regime by definition of the new variables. On the other hand, we focus on the Finsler geometry in order to find a Finslerian function which is equivalent with the mentioned non-commutative Lagrangian under special conditions. Also, the effective potential of system was calculated as a part of the corresponding conditions.

On L-reducible spherically symmetric Finsler metrics

Cumulative index of volumes 21–30

A Finsler manifold is said to be geodesically reversible if the reversed curve of any geodesic remains a geometrical geodesic. Well-known examples of geodesically reversible Finsler metrics are Randers metrics with closed [Formula: see text]-forms. Another family of well-known examples are projectively flat Finsler metrics on the [Formula: see text]-sphere that have constant positive curvature. In this paper, we prove some geometrical and dynamical characterizations of geodesically reversible Finsler metrics, and we prove several rigidity results about a family of the so-called Randers-type Finsler metrics. One of our results is as follows: let [Formula: see text] be a Riemannian–Finsler metric on a closed surface [Formula: see text], and [Formula: see text] be a small antisymmetric potential on [Formula: see text] that is a natural generalization of [Formula: see text]-form (see Sec. 1). If the Randers-type Finsler metric [Formula: see text] is geodesically reversible, and the geodesic flow of [Formula: see text] is topologically transitive, then we prove that [Formula: see text] must be a closed [Formula: see text]-form. We also prove that this rigidity result is not true for the family of projectively flat Finsler metrics on the [Formula: see text]-sphere of constant positive curvature.

We investigate a class of complex Finsler metrics on a domain D ⊂ ℂ n . Necessary and sufficient conditions for these metrics to be strongly pseudoconvex complex Finsler metrics, or complex Berwald metrics, are given. The complex Berwald metrics constructed in this paper are neither trivial Hermitian metrics nor conformal changes of complex Minkowski metrics. We give a characterization of complex Berwald metrics which are of isotropic holomorphic curvatures, and also give characterizations of complex Finsler metrics of this class to be Kähler Finsler or weakly Kähler Finsler metrics. Moreover, in the strongly convex case, we give characterizations of complex Finsler metrics of this class to be projectively flat Finsler metrics or dually flat Finsler metrics.

Chapter VI Finslerian manifolds of constant sectional curvature [4

Reversible Finsler metrics of constant flag curvature

We show that the space of (reversible) Finsler metrics on the two-torus 𝕋 2 whose geodesic flow is conjugate to the geodesic flow of a flat Finsler metric strongly deformation retracts to the space of flat Finsler metrics with respect to the uniform convergence topology. Along the proof, we also show that two Finsler metrics on 𝕋 2 without conjugate points, whose Heber foliations are smooth and with the same marked length spectrum, have conjugate geodesic flows.

In this chapter we study homogeneous Finsler spaces. In Sect. 4.1, we define the notions of Minkowski Lie pairs and Minkowski Lie algebras to give an algebraic description of invariant Finsler metrics on homogeneous manifolds and bi-invariant Finsler metrics on Lie groups. Then in Sect. 4.2, we present a sufficient and necessary condition for a coset space to have invariant non-Riemannian Finsler metrics. In Sect. 4.3, we study homogeneous Finsler spaces of negative curvature and prove that every homogeneous Finsler space with nonpositive flag curvature and negative Ricci scalar must be simply connected. In Sect. 4.4, we apply our result to study the degree of symmetry of closed manifolds. In particular, we prove that if a closed manifold is not diffeomorphic to a rank-one Riemannian symmetric space, then its degree of symmetry can be realized by a non-Riemannian Finsler metric. Finally, in Sect. 4.5, we study fourth-root homogeneous Finsler metrics. As an explicit example, we give a classification of all invariant fourth-root Finsler metrics on Grassmannian manifolds.