[Paper Review] On total flexibility of local structures of Finsler tori without conjugate points

Dong Chen|arXiv (Cornell University)|Oct 28, 2013

Advanced Differential Geometry Research5 references3 citations

TL;DR

This paper establishes that any local Finsler structure on a 2-dimensional manifold can be isometrically embedded into a Finsler torus without conjugate points, demonstrating the absence of local obstructions to such embeddings. The key method involves constructing a perturbed enveloping function from Busemann-type functions, which recovers the Finsler metric and ensures the resulting torus metric is smooth and conjugate-point-free, even preserving symmetry if the original metric is symmetric.

ABSTRACT

We show that given a point on a Finsler surface, one can always find a neighborhood of the point and isometrically embed this neighborhood into a Finsler torus without conjugate points.

Motivation & Objective

• To investigate whether local Finsler structures on surfaces can be embedded into globally defined Finsler tori without conjugate points.
• To determine if there are intrinsic local obstructions to constructing such embeddings in the Finsler setting, contrasting with the rigid Riemannian case.
• To extend the concept of Busemann functions to a more general, ray-independent enveloping function framework that captures the Finsler metric.
• To demonstrate that the absence of conjugate points imposes no local restrictions on Finsler metrics, even in non-symmetric cases.
• To preserve symmetry of the original metric in the embedding when the initial Finsler structure is symmetric.

Proposed method

• Define a $ C^k $-smooth enveloping function $ F $ on $ \partial D_\epsilon \times D_\epsilon $ using distance functions relative to geodesics through a base point.
• Construct a reference enveloping function $ F^0 $ for a constant Finsler metric $ \varphi_0 $ on $ \mathbb{R}^2 $, which serves as a model for perturbation.
• Perturb $ F^0 $ to a new function $ \tilde{F} $ on $ \partial D_\epsilon \times \mathbb{R}^2 $ via a partition of unity, ensuring $ \tilde{F} $ is a $ C^k $-small perturbation.
• Use Lemma 1 to show that $ \tilde{F} $ is an enveloping function for a $ C^k $ Finsler metric $ \tilde{\varphi} $ on $ \mathbb{R}^2 $, which agrees with the original metric $ \varphi $ on $ D_\epsilon $ and with $ \varphi_0 $ outside a larger disk.
• Construct a symmetric version of $ \tilde{F} $ when the original metric is symmetric, using the antisymmetry condition $ F(p,x) = -F(-p,x) $, and define a symmetric distance function $ \tilde{d}(x,y) = \max_p \tilde{F}(p,x) - \tilde{F}(p,y) $.
• Define the Finsler metric $ \tilde{\varphi}(x,v) $ as the directional derivative of $ \tilde{d} $, ensuring $ \tilde{\varphi} $ is $ C^k $ and conjugate-point-free via Lemma 2.

Experimental results

Research questions

• RQ1 Can every local $ C^k $ Finsler structure on a surface be isometrically embedded into a Finsler torus without conjugate points?
• RQ2 Does the absence of conjugate points in a Finsler torus impose any local restrictions on the possible Finsler metrics?
• RQ3 Can the construction of such embeddings preserve the symmetry of the original Finsler metric?
• RQ4 Is there a generalization of Busemann functions that allows for a global reconstruction of the Finsler metric without relying on a specific geodesic ray?
• RQ5 Can the geodesic flow of a Finsler torus without conjugate points be smoothly conjugate to that of a flat Finsler torus?

Key findings

• Any $ C^k $ Finsler surface metric with $ k \geq 3 $ admits a local isometric embedding into a $ C^k $ Finsler torus without conjugate points.
• The construction relies on perturbing a reference enveloping function associated with a constant Finsler metric, ensuring the resulting metric is $ C^k $-smooth and conjugate-point-free.
• The resulting Finsler torus metric $ \tilde{\varphi} $ agrees with the original metric $ \varphi $ on a neighborhood of the point $ p_0 $, preserving local geometry.
• If the original Finsler structure is symmetric, the constructed embedding can be made to preserve symmetry, resulting in a symmetric $ \tilde{\varphi} $.
• The method ensures that all geodesics in the resulting torus are minimal, which implies the absence of conjugate points via Lemma 2.
• The enveloping function framework allows for a ray-independent reconstruction of the Finsler metric, generalizing the role of Busemann functions.