Geometry of reflective submanifolds in Riemannian symmetric spaces

Abstract

The set $ℛ\left(B\right)$ of submanifolds conjugate to a given reflective submanifold $B$ in a Riemannian symmetric space $M$ has a structure of symmetric space. Using this structure, for a submanifold $N$ in $M$ we establish integral formulae which represent the integrals of the functions $C\mapsto \text{vol}(N\cap C)$ on $ℛ\left(B\right)$ by some extrinsic geometric amounts of $N$.