On the symmetry of Riemannian manifolds (Q2842809)

After the success of Riemannian symmetric spaces, various weaker notions of ``symmetric manifolds'' were introduced and studied. For example, A. Selberg introduced the notion of a weakly symmetric space as follows: A Riemannian manifold \((M,g)\) is called weakly symmetric if for any point \(x\in M\) and any vector \(\zeta\in T_xM\) there is some isometric involution \(\sigma:M\rightarrow M\) such that \(\sigma(x)=x\) and \(d\sigma|_{T_{x}M}(\zeta)=-\zeta\). Note that \(d\sigma|_{T_{x}M}\) need not be equivalent to \(-\mathrm{Id}\) on \(T_{x}M\). It is now a direct generalization to investigate what happens if one strenghens the conditions as follows:NEWLINENEWLINEA Riemannian manifold \((M,g)\) is called \(k\)-fold symmetric if for any point \(x\in M\) and any \(k\)-tuple of vectors \(\zeta_1, \dots, \zeta_k\in T_xM\) there is some isometric involution \(\sigma:M\rightarrow M\) such that \(\sigma(x)=x\) and \(d\sigma|_x(\zeta_i)=-\zeta_i\) for \(i=1, \dots, k\).NEWLINENEWLINEThe main theorem of the paper states now: A connected, simply connected 2-fold symmetric Riemannian manifold must be globally symmetric.NEWLINENEWLINEIn the last section, the author describes a generalization to Finsler spaces.