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Structure of geodesics for Finsler metrics arising from Riemannian g.o. metrics

Teresa Arias-Marco, Zdenek Dusek

Abstract

Homogeneous geodesics of homogeneous Finsler metrics derived from two or more Riemannian geodesic orbit metrics are investigated. For a broad newly defined family of positively related Riemannian geodesic orbit metrics, geodesic lemma is proved and it is shown that the derived Finsler metrics have also geodesic orbit property. These Finsler metrics belong to the newly defined class of the $α_i$-type metrics which includes in particular the $(α_1,α_2)$ metrics. Geodesic graph for the sphere ${\mathrm{S}}^7={\mathrm{Sp(2)}}{\mathrm{U}}(1)/{\mathrm{Sp(1)}{\mathrm{diag}}{\mathrm{U}}(1)}$ with geodesic orbit Finsler metrics of the new type $(α_1,α_2,α_3)$, arising from two or more Riemannian geodesic orbit metrics, is analyzed in detail. This type of metrics on $S^7$ is one of the missing cases in a previously published classification of geodesic orbit metrics on spheres.

Structure of geodesics for Finsler metrics arising from Riemannian g.o. metrics

Abstract

Homogeneous geodesics of homogeneous Finsler metrics derived from two or more Riemannian geodesic orbit metrics are investigated. For a broad newly defined family of positively related Riemannian geodesic orbit metrics, geodesic lemma is proved and it is shown that the derived Finsler metrics have also geodesic orbit property. These Finsler metrics belong to the newly defined class of the -type metrics which includes in particular the metrics. Geodesic graph for the sphere with geodesic orbit Finsler metrics of the new type , arising from two or more Riemannian geodesic orbit metrics, is analyzed in detail. This type of metrics on is one of the missing cases in a previously published classification of geodesic orbit metrics on spheres.

Paper Structure

This paper contains 4 sections, 4 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Example $S^{7}={\mathrm{Sp}}(2)\cdot{\mathrm{U}}(1)/{\mathrm{Sp}}(1)\cdot{\mathrm{diag}}({\mathrm{U}}(1))$

Key Result

Lemma \oldthetheorem

Let $(G/H,F)$ be a homogeneous Finsler space with a reductive decomposition ${\mathfrak{g}}={\mathfrak{m}}+{\mathfrak{h}}$. A nonzero vector $y\in{{\mathfrak{g}}}$ is geodesic vector if and only if it holds where the subscript ${\mathfrak{m}}$ indicates the projection of a vector from ${\mathfrak{g}}$ to ${\mathfrak{m}}$.

Theorems & Definitions (7)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Conjecture \oldthetheorem
  • Lemma \oldthetheorem
  • Theorem \oldthetheorem