Einstein hypersurfaces in irreducible symmetric spaces

Abstract

We give a full classification of Einstein hypersurfaces in irreducible Riemannian symmetric spaces of rank greater than 1 (the classification in the rank-one case was previously known). There are two classes of such hypersurfaces. The first class consists of codimension one Einstein solvmanifolds in irreducible symmetric spaces of noncompact type constructed via the Iwasawa decomposition. The second class consists of two exceptional families in low-dimensional symmetric spaces \({\overline{M}}=\text {SU}(3)/\text {SO}(3)\) and \({\overline{M}}=\text {SL}(3)/\text {SO}(3)\). Any Einstein hypersurface M in such space \({\overline{M}}\) is developable: it is foliated by totally geodesic spheres (respectively, by totally geodesic hyperbolic planes) of \({\overline{M}}\), with the space of leaves being a special Legendrian surface in \(S^5\) (respectively, a proper affine sphere in a \({\mathbb {R}}^3\)).