Totally geodesic hypersurfaces of four-dimensional generalized symmetric spaces

Abstract

We prove that a four-dimensional generalized symmetric space does not admit any non-degenerate hypersurfaces with parallel second fundamental form, in particular non-degenerate totally geodesic hypersurfaces, unless it is locally symmetric. However, spaces which are known as generalized symmetric spaces of type C do admit non-degenerate parallel hypersurfaces and we verify that they are indeed symmetric. We also give a complete and explicit classification of all non-degenerate totally geodesic hypersurfaces of spaces of this type.