Clifford-Wolf translations of left invariant Randers metrics on compact Lie groups (Q2922413)

An isometry of a homogeneous Riemannian manifold \((M,g)\) is called a Clifford-Wolf translation (CW-translation) if it moves all points of \(M\) the same distance. If, for any two points \(x_1\), \(x_2\in M\), there is a CW-translation \(\rho\) such that \(\rho(x_1)=x_2\), then \((M,g)\) is called Clifford-Wolf homogeneous. In [Comment. Math. Helv. 37, 65--101 (1962; Zbl 0113.15302)], \textit{J.~C.~Wolf\,} determined all CW-translations of a given Riemannian manifold. If, on a manifold \(M\), there is a continuous function \(F:TM\to\mathbb R^+\) satisfying (i)\,\(F(x,y)>0\) for any \(y\neq 0\), (ii)\,\(F(x,\lambda y)=\lambda F(x,y)\) for any \(y\in TM_x\) and \(\lambda>0\), (iii)\,the Hessian matrix \(g_{ij}=\frac12[F^2]_{y^iy^j}\) is positively definite, then \((M,F)\) is a Finsler space and \(F\) is a Finsler metric.NEWLINENEWLINENEWLINEAn important example of a Finsler metric is the Randers metric of the form \(F=\alpha+\beta\), where \(\alpha\) is a Riemannian metric and \(\beta\) is a 1-form. If \(d\) is a distance function of a Finsler space \((M,F)\), then an isometry \(\rho\) of \((M,F)\) such that \(d(x,\rho(x))\) is a constant function, is called a CW-translation of \((M,F)\).NEWLINENEWLINEIn this paper, the authors study CW-translations of Finsler spaces. They find the condition for a homogeneous Randers space to be CW-homogeneous and classify all CW-homogeneous Randers spaces. It is shown that if \(G\) is a compact, connected, simple Lie group and \(F\) is a left invariant Randers metric, then \((G,F)\) is CW-homogeneous if and only if the indicatrix of \(F\) in \(\mathfrak{g}\) is a round sphere with respect to a bi-invariant metric. A number of examples of non-reversible Finsler metrics which are CW-homogeneous is presented.