Titles and Abstracts

(Partial)

Marcelo Almeida de Souza ( msouza@mat.ufg.br )

Abstract: We consider Finsler spaces with a special Randers metric $F=\alpha+ \beta$, on the three dimensional real vector space, where $\alpha$ is the Euclidean metric and $\beta=bdx_3$ is a 1-form with norm $b,\, 0\leq b<1$. By using the notion of mean curvature for immersions in Finsler spaces introduced by Z. Shen, we get the ordinary differential equation that characterizes the helicoidal minimal surfaces with $x_3-$axis. We prove that for every $b,\, 0\leq b<1$, the classical helicoid is a minimal surface. We also prove that for every $b,\, 0< b<1$, minimal surfaces of rotation around a $l-$axis, $l \neq x_3$, do not exist. In an earlier paper we have studied the case $l=x_3$, where we obtained some results on the existence of complete minimal surfaces of rotation around the $x_3-$axis.

"On Chern-Finsler connection".

Vladimir Balan ( University Politehnica of Bucharest, Romania, vbalan@mathem.pub.ro

"Geodesics, paths and Jacobi fields in Finslerian jet models".

Abstract:

Within the framework of multi-parametric first-order Finslerian jet models, an adjusted version of the two arc-length variations is presented, and the Jacobi field equations are derived. Conjugacy and the effect of flag-curvature are discussed. The developed theory is further applied for the case of distinguished paths; attached illustrative Maple simulations for Finslerian 1-parametric jet models are provided.
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David Bao  ( University of Houston, U.S.A., bao@math.uh.edu )

"Some Remarks on Ricci Curvature".

Sandor Basco ( University of Debrecen, Hungary, bacsos@math.klte.hu )

"On a problem of Matsumoto and Shen"

The purpose of the present lecture is to discuss the following two problems:

Matsumoto's problem: "The most important problem on projectively Berwald spaces is, of course, to find the tensorial characterization of such spaces" Shen problem: Is there any Douglas metric which is not locally projectively Berwald.

Finally we give an example for Douglas space and Berwald space which is not locally common geodesic, and we would like to present a possible solution for the Shen problem.

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"A Classification of Berwald-type connections and its applications"

Robert Bryant (Duke University, U.S.A.,  bryant@math.duke.edu )

"Complex geometry and Finsler metrics with positive constant flag curvature"

I will show that a complete Finsler n-manifold with constant positive flag curvature defines a Kahler structure on its space of (oriented) geodesics. Moreover, the underlying complex structure on this space makes it biholomorphic to the complex quadric of dimension 2n-2. The Kahler form satisfies some extra identities that make the geometry of these Kahler manifolds particularly interesting. Finally, I explain how the recent results of Lebrun and Mason can be coupled with these results to yield a classification of the reversible Finsler metrics on the 2-sphere that have Finsler-Gauss curvature K = 1.

Shenglin Cao ( Beijing Normal University Department of Astronomy,100875 caosl20@yahoo.com.cn ):

"The Physics and Finsler Space-time".

In this talk, I will show that the relation between the physics and the space-time. Usually, most people would find Einstein's theory of relativity is only the theory about the space-time but the quantum theory not. But we know that, it is not space-time that is there and that impresses its form on things, but the things and their physical laws that determine space-time. So, if the quantum mechanics is a correct physical theory, then it will determine a structure of the space-time. This paper will describe the character of the structure of the Finsler space-time.

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Xinyue Chen ( Chongqing Institute of Technology, P.R. China, chenxy58@tom.com  chengxy@cqit.edu.cn):

The Douglas curvature D of Finsler metrics is an important non-Riemannian projective invariant constructed from the Berwald curvature. A Finsler metric is called  a Douglas metric if its Douglas curvature D. The Douglas metrics are more generalized ones than Berwald metrics and the class of Douglas metrics is much larger than that of Berwald metrics. Furthermore, the class of Douglas metrics includes the locally projectively flat Finsler metrics. The study on Douglas metrics will enhance our understanding of the geometric meaning of non-Riemannian quantities.
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In 2001, the authors proved that a Randers metric $F=\alpha +\beta$ with $\beta$
closed has isotropic mean Berwald curvature if and only if it has relatively isotropic
Landsberg curvature. We all know that a Randers metric $F=\alpha +\beta$ is a Douglas
metric if $\beta$ is closed. In this lecture, we discuss the Douglas metrics and some
of their important non-Riemannian curvature properties. Particularly, we generalize the
result as above and get the following
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Main theorem. Let F be a non-Riemannian Finsler metric on a manifold of
dimension $n\geq 3$. The following are equivalent:
(a) F is of isotropic Berwald curvature;
(b) F is a Douglas metric with isotropic mean Berwald curvature;
(c) F is a Douglas metric with relatively isotropic Landsberg curvature.
The projectively flat Finsler metrics are just special Douglas metrics. The authors
have completely classified projectively flat Randers metrics of almost isotropic S-curvature
two years ago. A natural problem is whether or not there are other types of projectively
flat Finsler metrics of almost isotropic S-curvature. In this lecture, we characterize
projectively flat Finsler metrics with almost isotropic S-curvature and completely
classify these metrics.

Irena Comic  ( Faculty of Technical Sciences,Serbia, comirena@uns.ns.ac.yu )

Vincze Csaba ( University of Debrecen,Hungary,  csvincze@math.klte.hu )

"On Matsumoto's problem of conformally equivalent Berwald manifolds".

Abstract: The original problem is that whether there exist conformally equivalent Berwald, or locally Minkowski manifolds. In this note we solve this problem proving that the scale function between such special Finsler manifolds must be constant unless they are Riemannian. As a direct consequence the uniqueness of the Wagner structure on a (non-Riemannian) Finsler manifold follows immediately - it is well-known due to M. Hashiguchi and Y. Ichijy\~{o} that a Finsler manifold is a Wagner manifold if and only if it is conformal to a Berwald manifold.
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Lilia Del Riego  ( lilia@galia.fc.uaslp.mx  )

"Semi-Riemannian themes in Finsler geometry: Geodesic Completness".

Abstract: It is well known that the Hopf-Rinow theorem guarantees the equivalence of geodesic completeness and metric completeness for arbitrary Riemannian manifolds. Further, either of these conditions implies the existence of minimal geodesics. The situation is very different for semi-Riemannian manifolds, much more interesting. I will present several important examples including both completeness conditions and the stability of completeness in semi-Riemaniann manifolds.

Patrick Foulon  (Institut de Recherche Mathematique Avancee, France, foulon@math.u-strasbg.fr )

TBA

Gabrijela Grujic  (Faculty of Technical Sciences, Serbia, gabrijela@neobee.net)

The subspaces in Miron's Osc^kM

Qun He  ( qun_he@163.com )

"On Bernstein Type Theorems in Finsler Spaces"

Abstract:By using the volume form induced from the projective sphere bundle of the Finsler manifold, we study properties of Finsler minimal submanifolds and establish the Bernstein type theorems for Finsler minimal graphs the Minkowski space and the Randers space. Firstly, we prove that the volume form for the Randers metric F = a+b in a Randers space is just that for the Riemannian metric a. If a is Euclidean, F = a+b is called a special Randers metric. Hence, by the Bernstein theorem for Euclidean minimal graphs, it follows that any complete minimal graphs in the special Randers m-space with m<8 are affine hyperplanes. Secondly, we consider hypersurfaces in the Minkowski space. It is proved that a constant mean curvature graph in the Minkowski space satisfies the so-called elliptic equation of mean curvature type. Therefore, any complete minimal graphs in a 3-dimensional Minkowski space are planes.

Laszlo Kozma ( University of Debrecen,Hungary,  kozma@math.klte.hu
):

"Hyperbolicity in Finsler geometry."

Abstract: The hyperbolicity of a Finsler manifold can be measured analytically by the flag curvature. Geometrically, there are several notions of hyperbolicity, or, of non-positive curvature, defined by Alexandrov, Gromov, Busemann, Petersen, etc. for (geodesic) metric spaces. We intend to study the relationships between them for Finsler manifolds. It is known that if a Finsler space is of non-positive curvature in the sense of Alexandrov, then the space is Riemannian. It is not the case for Busemann's condition. We show that for Berwald manifolds the non-positivity of the flag curvature is equivalent to Busemann's and Petersen's nonpositivity conditions.
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Nany Lee ( nany@uos.ac.kr ):

"On the connections on almost complex Finsler manifolds."

Abstract: For a Rizza manifold (M,J,L) we show that there exists a unique connection $\nabla\,$
such that $\nabla G=0\,$ and $\nabla J=0\,$ and its horizontal torsion is in a special form. Here $G\,$ is the generalized Finsler metric induced by $L\,.$ This is a generalization of the canonical connection of almost Hermitian manifold introduced by S. Kobayashi.

From its curvature, we define a new invariant, a holomorphic sectional curvature and we show that this holomorphic sectional curvature of an almost complex submanifold $M^\prime\,$ of $M\,$ is less than or equal to that of $M\,.$ This fact implies that a Rizza manifold is hyperbolic if its holomorphic sectional curvature is bounded above by -1.
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Xiaohuan Mo ( Peking University, P.R. China, moxh@pku.edu.cn ):

TBA

"A Sphere Theorem for Non-Reversible Finsler Metrics"

Abstract: For a Finsler metric F on a compact smooth manifold M we introduce the reversibility . If the manifold is simply-connected and the flag curvature satisfies , then the length of a closed geodesic is bounded from below by . This result implies that the manifold is homotopy-equivalent to the $n$-sphere and can be used to derive existence results for closed geodesics.
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Colleen Robles (University of Rochester, robles@math.rochester.edu ):

TBA

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Yiming Long (Nankai Institute of Mathematics,longym@nankai.edu.cn ):

"Multiple closed geodesics on Finsler 2-spheres"

In a recent joint paper of V. Bangert and Y. Long, the following result is proved: For any Finsler metric on the 2 dimensional sphere, there exist always at least two prime closed geodesics. This result gives a positive answer for $S^2$ to a conjecture of D. V. Anosov proposed in 1974 based on the example of A. Katok.

Yi-Bing Shen  ( Zhejiang University, yibingshen@zju.edu.cn  )

"On Variation Calculus in Finsler Geometry "

In this talk, I would like to give a report on our recent works on variation calculus of harmonic maps and minimal immersions in Finsler geometry, jointed with Qun He and Yan Zhang, respectively. By using the volume form induced from the projective sphere bundle over Finsler manifolds, the first and second variation formulas for the nondegenerate map between Finsler manifolds are derived. The volume variation formula for Finsler submanifolds is also derived. We then show that an isometric immersion of Finsler manifolds is harmonic if and only if it is minimal. Some interesting results on harmonic maps and minimal immersions of Finsler manifolds are given.

Zhongmin Shen  (Indiana University - Purdue University Indianapolis, zshen@math.iupui.edu )

"Finsler Manifolds with isotropic S-curvature".

In this talk, I will discuss some local and global rigidity results on Finsler manifolds with zero, constant or isotropic S-curvature.

Jelena Stojanov  (Technical Faculty "MihajloPupin", Serbia and Montenegro, jelena@tf.zr.ac.yu )

The Spray Theory in the subspaces of Miron's Osc^kM

L. Tamassy

(Debrecen University, Hungarytamassy@math.klte.hu)

"Affine deformation of locally Minkowski spaces".

Abstract: Let $\ell\Cal M^n=(M,\Cal F)$ be a locally Minkowski space, $A$ a field of affine transformations on $M$, $a(p):T_pM\to T_p M$, $a\in A$, $p\in M$, and $\Cal I(p_0):=\{y\in T_{p_0}M\mid \Cal F(p_0,y)=1\}$ the indicatrix at $p$. Then $\{\tilde{\Cal I} (p):=a(p)\Cal I(p)\}$ yields a Finsler space $A(\ell\Cal M^n)=\tilde{F}^n=(M,\tilde{\Cal F})\equiv (M,\tilde{\Cal I})$, which is no longer Minkowskian. We investigated these spaces ([4]--[7]) in the case if $a(p)$ are centroaffine transformations, and showed that over $M$ with an affine structure exactly these spaces admit metrical linear connections in $TM$. The problem is also related to the Finsler metrizability of a linear connection investigated by M. Anastasiei [2], [3]. The case, when $a(p)$ are affine, but not centroaffine was considered first by D. Hrimiuc [3]. Randers spaces belong to this family. We want to show that these spaces admit interesting metrical connections in $TM$. They can not be linear, yet they are very near to the linear connections (they could be called quasi-linear).

We investigate the geometry and the structure of these affinely deformated locally Minkowski spaces.

Keti Tenenblat

Abstract: We consider Finsler spaces with a Randers metric $F = \alpha + \beta$, on the three dimensional real vector space, where $\alpha$ is the Euclidean metric and $\beta$ is a 1-form with norm $b,\,\,0\leq b<1$. By using the notion of mean curvature for immersions in Finsler spaces, introduced by Z. Shen, we obtain the partial differential equation that characterizes the minimal surfaces which are graphs of functions. For each b, $0\leq b< 1/\sqrt{3}$, we prove that it is an elliptic equation of mean curvature type. Then the Bernstein type theorem and other properties, such as the nonexistence of isolated singularities, of the solutions of this equation follow from the theory developed by L. Simon. For $b\geq 1/\sqrt{3}$,the differential equation is not elliptic. Moreover, for every $b,\,\,1/\sqrt{3}<b<1$ we provide solutions, which describe minimal cones, with an isolated singularity at the origin.