Orbits of \(s\)-representations (Q2716804)

A Riemannian manifold \(M\) is called \(k\)-symmetric if for each \(x\in M\) there is an isometry \(S_x\) such that (i) the order of \(S_x\) is \(k\); (ii) \(x\) is an isolated fixed point of \(S_x\); (iii) \(S_x\circ S_y\circ S_x^{-1}=S_z\), where \(z=S_x(y)\). Particularly, 2-symmetric spaces are symmetric spaces. Orbits of the isotropy representations of semi-simple Riemannian symmetric spaces \(M\) are investigated. These are called orbits of \(S\)-representations or \(R\)-spaces. If the isometric immersion \(f:M\to \mathbb{R}^{m +d}\) has a pointwise 3-planar normal section, then it is said to satisfy the P3-PNS property. Necessary conditions for the natural imbedding \(f:M\to \mathbb{R}^{m+d}\) of an \(R\)-space to have this property are given.