Symmetric submanifolds in symmetric spaces.
A submanifold \(N\) of a Riemannian manifold \(M\) is called a symmetric submanifold if for each point \(p\) in \(N\) there exists an involutive isometry of \(M\) which fixes \(p\), leaves \(N\) invariant and whose differential at \(p\) fixes the normal vectors of \(N\) at \(p\) and reflects the tangent vectors. (For \(M= E^n\), see \textit{D. Ferus}, [Math. Ann. 247, 81--93 (1980; Zbl 0446.53041)].) A submanifold \(N\) of \(M\) is called reflective if the geodesic reflection of \(M\) in \(N\) is a well defined global isometry [see \textit{D. S. P. Leung}, J. Differ. Geom. 14, 167--177, 179--185 (1979; Zbl 0453.53044).] Let \(M\) be an irreducible symmetric space of non-compact type with rank greater than one. Let \(T\) be a reflective submanifold of \(M\) such that the Euclidean factor in the de Rham decomposition of \(T\) is one-dimensional. Let \(o\in T\), denote by \(\gamma\) the exponential image of the one-dimensional component of \(T\) at \(o\) and assume that the geodesic \(\gamma\) is parametrized by arc length so that \(o= \gamma(0)\). Let \(Q\) be the reflective submanifold of \(M\) perpendicular to \(T\) at the point \(o\), and \(G_Q\) the Lie subgroup of the isometry group of \(M\), which is generated by the symmetries of \(Q\). It is proved that then \(G_Q\cdot o= Q\), and the orbit of \(G_Q\) through any point on \(\gamma\) different from \(o\) is a non-totally geodesic symmetric submanifold of \(M\). A list of such triples \((M,T,Q)\) is provided in a table.
scientific article; zbMATH DE number 1780870
scientific article