The 56-th Symposium on Finsler Geometry

 September 2–5, 2025 Sapporo

September 3 (Wednesday) First day

Registration at Tokai University: 9:10~9:45

Opening ceremony and announcement from the organizer : 9:50~10:00　              

10:00~11:25: Chairman TBA

1. 10:00~11:00 Xinyue CHENG, Isoperimetric inequality on Finsler metric measure manifolds with integral weighted Ricci curvature bounds

Abstract.

The isoperimetric inequality has always been one of the most important topics in geometry and has attracted extensive attention. In this talk, we carry out in-depth research centering around the isoperimetric inequality on Finsler metric measure manifolds with integral weighted Ricci curvature bounds. For our aim, we first establish Laplacian comparison theorem, Bishop-Gromov type volume comparison theorem and relative volume comparison theorem on such Finsler manifolds. Based on these, we prove a local Dirichlet isoperimetric constant estimate under the integral weighted Ricci curvature bounds. As applications of the Dirichlet isoperimetric constant estimate, we get first Dirichlet eigenvalue estimate and a gradient estimate for harmonic functions.

2. 11:05~11:25 Rattanasak HAMA and Sorin V. SABAU, The geometry of Randers cylinders of revolution with non-constant

                                                                                                                                                  navigation data along meridians

Abstract.

We study the structure of cut loci of a Finsler metric of Randers type defined on a cylindrical surface of revolution. Our Randers metrics are obtained by perturbing the Riemannian metric with a closed one-form, which is equivalent to the solution of Zermelo’s navigation problem for a wind blowing along meridians. We describe the Riemannian part by a solution of a Riccati type differential equation approach that allows a better control of cut loci and construction of examples. We establish the conditions for the Finsler metric to have the same cut locus structure as in the Riemannian case, giving several examples.

11:25~11:35 Coffee break

11:35~11:55: Chairman TBA

3. 11:35~11:55 Laszlo KOZMA, On connections of sub-Finslerian geometry　

Abstract.

A sub-Finslerian manifold is, roughly speaking, a manifold endowed with a Finsler type metric which is defined on a k-dimensional smooth distribution only, not on the whole tangent manifold. First it is presented the construction of a generalized nonlinear connection for a sub-Finslerian manifold, called L-connection by the Legendre transformation which characterizes normal extremals of a sub-Finsler structure as geodesics of this connection. Then the results of the investigation of some of its properties like normal, adapted, partial and metrical are shown.

11:55~13:00   Lunch break

13:00~14:05: Chairman TBA

4. 13:00~13:30 Piotr KOPACZ, Extensions of Matsumoto slope-of-a-mountain problem - part I

Abstract.

In this talk the Matsumoto slope-of-a-mountain problem is revisited. By introducing the concept of a slippery slope and constructing new models for time-minimizing navigation on a mountain slope under action of gravity we generalize and interlink Matsumoto’s problem with Zermelo’s navigation problem on Riemannian manifolds. The purely geometric solutions for optimal navigation on a slippery mountain slope are discussed by means of Finsler geometry. Using some relevant examples the developed theory is illustrated, while explaining and comparing the deformations of the Finslerian indicatrices and the behavior of time geodesics with different approaches to the gravity effect. The talk is based on joint work with Nicoleta Aldea. 

5. 13:35~14:05 Nicoleta ALDEA, Extensions of Matsumoto slope-of-a-mountain problem - part II

Abstract.

By the anisotropic deformation of the background Riemannian metric and rigid translation with the use of the rescaled gravitational wind, we obtain the purely geometric solutions for time-minimizing navigation in the models of a slippery mountain slope, which are given by the new Finsler metrics of general $(\alpha, \beta)$-type. The related strong convexity conditions for the slippery slope metrics under which the related time-minimizing trajectories can be described as Finslerian geodesics are thoroughly established. This is joint work with Piotr Kopacz.

14:05~14:15: Coffee break

14:15~16:40: Chairman TBA

6.    14:15~14:55 Florent BALACHEF, Isosystolic Inequalitiess on two-dimensional tori

Abstract.

In this talk I will survey all known optimal isosystolic inequalities on two-dimensional Finsler tori involving the following two central notions of Finsler area: the Busemann-Hausdorff area and the Holmes-Thompson area. This is joint work with Teo Gil Moreno de mora i Sardà.

7.    15:00~15:40 Zoltan MUZSUNAY, New results on the holonomy of Finsler manifolds　

Abstract.

The holonomy group of a Riemannian or Finslerian manifold is generated by parallel translations along loops. While Riemannian holonomy is completely classified, the Finslerian case is far less understood.

We study the holonomy of two-dimensional simply connected Randers manifolds of constant curvature by computing their infinitesimal holonomy algebras. Our results show that the local holonomy can be either infinite-or finite-dimensional, revealing fundamental geometric differences despite their common origin in navigation data.

In the infinite-dimensional case, we prove that the holonomy is maximal, with closure isomorphic to the group of orientation-preserving

diffeomorphisms of the circle.

The finite-dimensional cases are particularly intriguing because, as our previous result shows, almost all Finsler manifolds have

infinite-dimensional holonomy, hence finite-dimensional holonomy is `rare'. We establish the first examples of Randers surfaces with non-trivial

finite-dimensional holonomy, which are neither Riemannian, Berwaldian, nor (singular) Landsbergian.

8.    15:45~16:05 Takayoshi OOTSUKA, Classical Unification Theory from Finsler geometry

Abstract.

We consider a unification model which combines gravity and electromagnetic force in terms of Finsler geometry. Though the Kaluza-Klein model is very famous for such as unification theory, we may give another unification one.

9.    16:10~ 16:40 Chunping ZHONG,   On holomorphic invariant Kahler-Finsler metrics

Abstract.

According to S.-S. Chern, Finsler geometry is Riemannian geometry without quadratic restriction. In this talk we are concerned with the problem of constructing holomorphic invariant Kahler-Finsler metrics which are not necessary Hermitian quadratic. We show that there exists no Aut(Bn)-invariant complex Finsler metric other than a positive constant multiple of the Bergman metric on the open unit ball Bn in Cn , while there exist infinitely many Aut(Pn)-invariant Kahler-Finsler metrics on the unit polydisk Pn in Cn with n ≥ 2, which are non-Hermitian quadratic. This phenomenon also happens on the irreducible bounded symmetric domains of type I-IV . We obtain a general Schwarz lemma on a classical domain D whenever D is endowed with an arbitrary Aut(D)-invariant Kahler-Finsler metric  F. Our results show that the Lu constant associated to (D, F) is both an analytic invariant and a geometric invariant which is better understood in complex Finsler setting.

September 4 (Thursday)  Second day

10:00~11:35: Chairman TBA

10. 10:00~11:00  Huagui DUAN, Multiplicity of closed geodesics on Finsler manifolds

Abstract.

In this talk, firstly we will introduce the Maslov-type theory and the generalized common index jump theorem. Then we will talk about its applications to the multiplicity of closed geodesics on Finsler manifolds.

11:00~11:10 Coffee break

11. 11:10~11:45 Adela MIHAI, Torqued and anti-torqued vector fields

Abstract.

We first study the existence of torqued and anti-torqued vector fields on the hyperbolic ambient space H^n. Although there are examples of proper torqued vector fields on open subsets on H^n, we prove that there is no a proper torqued vector field globally defined on H^n. Similarly,

we have another non-existence result for anti-torqued vector fields, as long as their conformal scalar is a non-constant function. Moreover,

when the conformal scalar is constant, some examples of anti-torqued vector fields are provided. In the second part of the talk we study

rectifying submanifolds of a Riemannian manifold endowed with an anti-torqued vector field. For this, we first determine a necessary and

sufficient condition for the ambient space to admit such a vector field. Then we characterize submanifolds for which an anti-torqued vector

field is always assumed to be tangent or normal. A similar characterization is also done in the case of the torqued vector fields. We obtain that

the rectifying submanifolds with anti-torqued axis are the warped products whose warping function is a first integration of the conformal

scalar of the axis. This is a joint work with Muhittin Evren AYDIN and Cihan OZGUR under the TUBITAK Project Grant 123F451.

11:45~13:00 Lunch break

13:00~14:55: Chairman TBA

12.   13:00~13:40 Benling LI, The nonlinearity of Sobolev spaces over Finsler manifolds

Abstract.

As is known, the Sobolev spaces over Riemannian manifolds are always linear. However, this phenomenon fails in the Finslerian framework. The first counterexample was found by A. Kristály and I.J. Rudas in 2014, which is the Sobolev space over a (special) Funk space. In this talk, we will present more various examples, i.e., the Sobolev spaces over all kinds of Funk spaces and Berwald spaces. Besides, we also find the connection between the linearity of Sobolev spaces and the reversibility/backward completeness of the base Finsler manifold. (This is a joint work with Wei Zhao).

13.    13:45~14:25 Qialing XIA, On almost square Ricci solitons

Abstract.

In this talk, I will introduce the almost Ricci solitons in Finsler geometry. In particular, we focus on almost square Ricci solitons $(M, F, V)$ defined by a square metric $F$ and a vector field $V$ on an $n$-dimensional manifold $M$.  We prove that $(M, F, V)$ is an almost square Ricci soliton if and only if $F$ is Ricci flat and $V$ is a conformal vector field of $F$ when $n\geq 2$, and it is an almost locally projectively flat square Ricci soliton if and only if $F$ is of zero flag curvature and $V$ is a Killing vector field of $F$ when $n\geq 3$. As applications, we determine the local structures of almost (resp. locally projectively flat) square Ricci solitons.

14:25~14:35 Coffee break

14:35~15:15: Chairman TBA

14. 14:35~15:15 Erico TANAKA, Parameter independent Lagrange formalism and Helmholtz conditions

Abstract.

Helmholtz conditions are the necessary and sufficient conditions for a given set of 2nd order ODEs to have a Lagrangian which then they become the Euler-Lagrange equations. In this talk, we discuss the Helmholtz conditions for a Finsler-like Lagrangian to exist in the setting of parameter invariant formalism of calculus of variation. 

Dinner Party 17:00 ~  University restaurant

September 5 (Friday)  Third day

10:00~12:30: Chairman TBA

15. 10:00~10:40 Wei ZHAO, On the Finslerian Laplacian

Abstract.

It is well-known that the Laplacian operator in the Finsler setting is nonlinear, representing a fundamental difference between Finsler geometry and Riemannian geometry. In this talk, I will present the spectral theory and  the heat flow associated with this operator

10:40~10:50 Coffee break

16. 10.50~11.50 Chunhui QIU, The Schwarz lemmas and their applications on complex Finsler manifolds 

Abstract.

The classical Schwarz lemma is one of the core contents of complex analysis and one of the most beautiful results in complex analysis. It has widespread applications and profound implications. Many famous mathematicians have explored the Schwarz lemmas and their applications from the perspectives of both function theory and differential geometry. In this talk, we generalize the Schwarz lemmas to complex Finsler manifolds. Furthermore, we give the Schwarz lemmas on complex Finsler manifolds with growing or decaying conditions.

11:50~13:00  Lunch break

September 6 (Saturday)  Free Day

Excursion: A visit Otaru City if you are interested.

Highschool students poster session

1. Chayapon CHAINARONG, The rotational Randers metric on surface of revolution

2. Natnicha RAKBUMRUNG, The Geodesic on rotational Randers surface of revolution

3. Chanon BAMPENG, The Clairaut relation on rotational Randers surface of revolution

4. Raika NINOMIYA and Shu SUDO, Visualising Trigonometric Formula -The Sum-to-Product and Product-to-Sum Formulae-