Clifford-Wolf homogeneous Randers spaces (Q2856391)

Let \((M, F)\) be a connected Finsler space and \(d\) its distance function. If \(F=\alpha+\beta\), where \(\alpha\) is a Riemannian metric and \(\beta\) is a smooth \(1\)-form on \(M\) whose length with respect to \(\alpha\) is everywhere less than \(1\), then \(F\) is called a Randers metric on \(M\). The Randers metric \(F\) can be uniquely written as \(F(x,y)=\lambda^{-1}\,\left(\sqrt{h(y,W)^2+\lambda\,h(y,y)}-h(y,W)\right)\), where \(h\) is a Riemannian metric, \(W\) is a vector field on \(M\) with \(h(W,W)<1\), and \(\lambda=1-h(W,W)\). The pair \((h,W)\) is called the navigation data of the Randers metric \(F\). An isometry \(\rho\) of \((M,F)\) is called a Clifford-Wolf translation if the function \(d(a,\rho(x))\) is constant on \(M\). A Finsler manifold \((M,F)\) is called Clifford-Wolf homogeneous if for any two points \(x,x'\in M\) there is a Clifford-Wolf translation \(\rho\) such that \(\rho(x)=x'\), and it is called restrictively Clifford-Wolf homogeneous if for any point \(x\in M\) there is a neighborhood \(V\) of \(x\) such that for any two points \(x_1, x_2\in V\) there is a Clifford-Wolf translation \(\rho\) such that \(\rho(x_1)=x_2\).NEWLINENEWLINEIn this paper, the authors give a complete classification of connected simply connected Clifford-Wolf homogeneous Randers spaces. They prove that a connected simply connected Randers space \((M,F)\) is Clifford-Wolf homogeneous if and only if \(M\) is a product of odd-dimensional spheres, connected simply connected compact simple Lie groups, and an Euclidean space. Also, they show that the Finsler metric \(F\) has the navigation data \((h, W)\) such that \((M,h)\) is Clifford-Wolf homogeneous, i.e., \(h\) is a Riemannian product of Riemannian metrics on spheres of constant curvature, bi-invariant metrics on Lie groups, and a flat metric on the Euclidean space with respect to the product decomposition of \(M\), and \(W\) is a Killing vector field of constant length with \(\|W\|_h<1\).