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On the symmetry of Riemannian manifolds. (English) Zbl 1273.53047
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:
A 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 \(i=1, \dots, k\).
The main theorem of the paper states now: A connected, simply connected 2-fold symmetric Riemannian manifold must be globally symmetric.
In the last section, the author describes a generalization to Finsler spaces.

MSC:
53C35 Differential geometry of symmetric spaces
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
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