List of Intended Presentation Titles and Abstracts of the Participants to the 55-th Symposium on Finsler geometry 2020, Japan

¡úXinyue Cheng (Chongqing Normal University, Chongqing, China)

Title:**Some important applications of improved Bochner inequality on Finsler manifolds**

Abstract: In this talk, we will i ntroduce the Bochner-Weitzenbock formula and the corresponding Bochner inequality on Finsler manifolds. As the applications, we will introduce some important applications of improved Bochner inequality on Finsler manifolds

¡ú **Salah Gomaa Elgendi **(Benha University - Egypt)

Title: Solutions for the Landsberg unicorn problem in Finsler geometry (20 min)

¡ú **Sandor Hajdu** (University of Antwerp, Belgium)

Title: Jacobi fields and conjugate points for a projective class of sprays (30-40 min)

Abstract. We investigate Jacobi fields and conjugate points in the context of sprays. We first prove that the conjugate points of a spray remain preserved under a projective change. Then, we establish conditions on the projective factor so that the projectively deformed spray meets the conditions of a proposition that ensures the existence of conjugate points. We use three one-parameter family of Finsler metrics as a running example throughout the talk. We will use one of these Finsler metrics to show how an analysis of cut and conjugate points can help us to reach a conclusion about conjugate points. All this is a joint work with Tom Mestdag (University of Antwerp).

¡ú** Rattanasak Hama** (Prince of Songkla University, Surat Thani Campus, Thailand)

Title: TBA

¡ú **Ioana Masca** (Saguna Highschool, Brasov, Romania)

Title: A Finsler metric of constant Gauss curvature K = 1 on 2-sphere

Abstract. We construct a concrete example of constant Gauss curvature K = 1 on the 2-sphere having all geodesics closed and of some length. The result is actually an (¦Á, ¦Â) Finsler metric, albeit a quite complicated one

¡ú **Zoltan Muzsnay** (University of Debrecen, Debrecen, Hungary)

Title: Almost all Finsler metrics have infinite dimensional holonomy group (40-60 min)

¡ú **Christian Pfeifer **(Bremen, Germany)

Title: The gravitating kinetic gas - Lifting the Einstein Vlasov system to the tangent bundle

Abstract: In this talk I will present a new model for the description of a gravitating kinetic gas, by coupling the 1-particle distribution function (1PDF) of the gas directly to the gravitational field, lifted to the tangent bundle of spacetime. This procedure takes the influence of the velocity distribution of the kinetic gas particles on their gravitational field fully into account, instead of only on average, as it is the case for the Einstein-Vlasov system.

By using Finsler spacetime geometry I construct an action for the kinetic gas on the tangent bundle, which is added as matter action to a canonical Finslerian generalisationof the Einstein-Hilbert action. The invariance of the kinetic gas action under coordinate changes gives rise to a new notion of energy-momentum conservation of a kinetic gas in terms of an energy-momentum distribution tensor. The variation of the total action with respect to the spacetime geometry defining Finsler Lagrangian yields a gravitational field equation on the tangent bundle, which determines the geometry of spacetime directly from the full non-averaged 1PDF. This equation can be regarded as generalisation of the Einstein-Vlasov system, which takes all features of the kinetic gas into account.

¡ú **S. V. Sabau**, Tokai University, Japan

Title: On the variational problem on Kropina manifolds

¡ú **Samaneh Saberali** (Urmia University, Iran)

Title: Concircular transformations in Finsler geometry (20 min)

¡ú** H. Shimada**, Tokai University, Japan

Title: On the Barthel connection

¡ú **Nicoleta Voicu** (Transylvania University, Brasov, Romania)

Title: A mathematical framework for Finsler extensions of Einstein gravity theory

(joint work with Christian Pfeifer and Manuel Hohmann).

**Abstract**: In modern field theory, fields are treated as sections of a certain fibered manifold (Y,¦Ð,X), called the configuration manifold, Lagrangians are treated as differential forms on some jet bundle J^{r}Y and variations of the action are expressed as Lie derivatives of the Lagrangian with respect to certain vector fields on J^{r}Y.

The present talk proposes a framework which allows us to apply the above machinery in constructing well-defined actions over Finsler spacetime manifolds. First, we construct configuration bundles whose sections are Finslerian geometric objects: Finsler functions, homogeneous d-tensors etc. Such configuration bundles sit over the positive (or oriented) projective tangent bundle of the spacetime manifold. Second, we show that general covariance of Lagrangians on such bundles leads to a notion of energy-momentum tensor which obeys an averaged conservation law.

A concrete vacuum action is then briefly discussed, starting from an argument by Pirani and using the so-called variational completion algorithm.

¡ú **Ming Xu** (Capital Normal University, Beijing, China)

Title: Laugwitz Conjecture and Landsberg Unicorn Conjecture in some special case, and applications for the isoparametric foliations on unit spheres (60 min)

Abstract. In a recent cowork with V. Matveev, we proved the Laugwitz Conjecture and Landsberg Unicorn Conjecture for Minkowski

norms with $SO(k)\times SO(n-k)$-symmetry. The Lie method and ODE method in our proofs inspire me to use an isoparametric foliation on the unit sphere to define and study the induced Minkowski norms and Hessian isometries.

¡ú **Mehran Gabrani** (Urmia University, Urmia, Iran)

Title: On Finsler warped product metrics of isotropic Berwald curvature

Random geodesic walks and Brownian motions on Finsler manifolds