[Paper Review] Geodesic orbit Finsler metrics on Euclidean spaces

Ming Xu, Shaoqiang Deng+1 more|arXiv (Cornell University)|Jul 9, 2018

Advanced Differential Geometry Research21 references3 citations

TL;DR

This paper investigates geodesic orbit Finsler metrics on Euclidean spaces, proving that such spaces fiber over symmetric Finsler spaces of non-compact type, with totally geodesic nilmanifold fibers of step-size at most 2. The key contribution is a structural decomposition of geodesic orbit Finsler metrics on Euclidean spaces via Finslerian submersions and Levi decomposition, with applications to curvature rigidity and new proofs of existing theorems on negative curvature conditions.

ABSTRACT

A Finsler space $(M,F)$ is called a geodesic orbit space if any geodesic of constant speed is the orbit of a one-parameter subgroup of isometries of $(M, F)$. In this paper, we study Finsler metrics on Euclidean spaces which are geodesic orbit metrics. We will show that, in this case $(M, F)$ is a fiber bundle over a symmetric Finsler space $M_1$ of non-compact type such that each fiber $M_2$ is a totally geodesic nilmanifold with a step-size at most 2, and the projection $π:M ightarrow M_1$ is a Finslerian submersion. Furthermore, when $M_1$ has no Hermitian symmetric factors, the fiber bundle description for $M$ can be strengthened to $M=M_1 imes M_2$ as coset spaces, such that each product factor is totally geodesic in $(M,F)$ and is a geodesic orbit Finsler space itself. Finally, we use the techniques in this paper to discuss the interaction between the geodesic orbit spaces and the negative (non-positive) curved conditions, and provide new proofs for some of our previous results.

Motivation & Objective

• To classify geodesic orbit Finsler metrics on manifolds diffeomorphic to Euclidean spaces.
• To understand the geometric structure of such metrics through fiber bundle decompositions and Finslerian submersions.
• To investigate the interaction between the geodesic orbit condition and curvature constraints, particularly negative or non-positive flag curvature.
• To provide new proofs for rigidity theorems on negatively curved geodesic orbit Finsler spaces and Berwaldian metrics.
• To correct and strengthen prior results on nilpotent step-size bounds in geodesic orbit Finsler nilmanifolds.

Proposed method

• Utilizes the Levi decomposition of the isometry algebra to analyze the interaction between geodesic orbit conditions and the nilradical structure.
• Applies Finslerian submersion techniques to decompose the total space as a fiber bundle over a symmetric Finsler space of non-compact type.
• Employs the spray vector field and reductive decomposition to verify total geodesy of fibers and product factors.
• Uses the $B_{rak{g}}$-orthogonal complement to define tangent spaces and analyze the action of isotropy subgroups.
• Applies curvature rigidity theorems from homogeneous Finsler geometry, particularly those involving Killing vector fields of constant length.
• Revisits and re-proves Theorems 6.2 and 6.3 on negative curvature and non-positive flag curvature using the new structural framework.

Experimental results

Research questions

• RQ1 What is the global geometric structure of a geodesic orbit Finsler metric on a Euclidean space?
• RQ2 How does the Levi decomposition of the isometry algebra interact with the geodesic orbit condition in Finsler geometry?
• RQ3 Under what conditions does a geodesic orbit Finsler space on a Euclidean space decompose as a product of totally geodesic symmetric and nilpotent factors?
• RQ4 Can the step-size of the nilradical in a geodesic orbit Finsler nilmanifold exceed 2?
• RQ5 What rigidity results emerge when combining the geodesic orbit condition with negative or non-positive flag curvature?

Key findings

• Any geodesic orbit Finsler metric on a Euclidean space arises as a Finslerian submersion over a symmetric Finsler space of non-compact type.
• Each fiber of the submersion is a totally geodesic nilmanifold with step-size at most 2.
• When the symmetric base space has no Hermitian symmetric factors, the total space is isometric to a product $M_1 \times M_2$ of totally geodesic geodesic orbit Finsler spaces.
• The nilradical of the isometry group has step-size at most 2, correcting a gap in prior work.
• A negatively curved geodesic orbit Finsler space must be a rank-one Riemannian symmetric space of non-compact type.
• A geodesic orbit Finsler space with non-positive flag curvature and negative Ricci scalar is symmetric and Berwaldian.