Geometry of reflective submanifolds in Riemannian symmetric spaces
Let \(M\) be a complete connected Riemannian manifold. A connected component of the fixed point set of an involutive isometry of \(M\) is called a reflective submanifold. Let \(\mathcal{R}(B)\) be the set of submanifolds conjugate to a given reflective submanifold \(B\) in a Riemannian symmetric space \(M\). The author proves that the set \(\mathcal{R}(B)\) has the structure of a symmetric space. In fact, if \(M\) is a Riemannian symmetric space of compact type, then so is \(\mathcal{R}(B)\). If \(M\) is a Riemannian symmetric space of noncompact type, \(\mathcal{R}(B)\) is a semi-Riemannian symmetric space of semisimple type. Using the structure of the symmetric space for a submanifold \(N\) in \(M\) the author establishes integral formulae which represent the integrals of the function \(C\rightarrow \text{vol}(N\cap C)\) on \(\mathcal{R}(B)\) by some extrinsic geometric quantities of \(N\).
scientific article; zbMATH DE number 5015998