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.

Tadashi Aikou (Kagoshima University)

"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.

**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.

**Behroz Bidabad**
(**Amirkabir university (Tehran Polytechnic), Tehran Iran,
bidabad@aut.ac.ir)**

**"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.

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

"**Douglas Metrics with Special Non-Riemannian
Curvature Properties"**

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.

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

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 )

"Hamilton Spaces of Higher Order"

**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.

**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.

**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.

**Xiaohuan Mo ( **
Peking University, P.R. China, ** moxh@pku.edu.cn** )**: **

**TBA**

**Hans-Bert Radamacher
**
**rademach@mathematik.uni-leipzig.de**** **

**"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.

**Colleen Robles (University
of Rochester,
robles@math.rochester.edu ):**

**TBA**

**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, Hungary**,
tamassy@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**

(**Universidade de Brasilia, Brazil**,
**keti@mat.unb.br **
)

**"A Bernstein type theorem on a Randers space". **

**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.