Symmetric spaces with rectangular unit lattices, revisited

Abstract

In this paper we study compact Riemannian symmetric spaces whose maximal tori are as simple as possible: metric products of circles. In particular, using representation theory, we construct explicitly an isometric embedding of such spaces $X$ into Euclidean space with the property that geodesic symmetries of $X$ extend to reflections along normal spaces. The converse is also true: extrinsically symmetric submanifolds have rectangular maximal tori. Further, such embedding of $X$ is unique up to congruence, and all isometries of $X$ extend to isometries of the ambient Euclidean space.