On the Geometry of Semi-Symmetric Hypersurfaces in Euclidean Space

Semi-symmetric spaces are Riemannian manifolds whose curvature tensor at a given point is the same as the curvature tensor of a symmetric space. The semi-symmetric spaces have been studied first by E. Cartan in connection with his research on locally symmetric spaces. In 1982, Z. Szabo gave a local classification of Riemannian semi-symmetric spaces dividing them into three classes: "trivial" class, consisting of all locally symmetric spaces and all two-dimensional Riemannian manifolds; "exceptional" class, consisting of all elliptic, hyperbolic, Euclidean and Kählerian cones; "typical" class, consisting of all Riemannian manifolds foliated by Euclidean leaves of codimension two. The "trivial" semi-symmetric spaces are well known and the "exceptional" ones are described and constructed explicitly by Z. Szabo.In this book, we study semi-symmetric hypersurfaces in the Euclidean n-space belonging to the "typical" class, considering them with respect to their second fundamental tensor. The main tool in our investigation is the system of Codazzi equations, which gives us a new approach to the classification, geometric description and construction of foliated semi-symmetric hypersurfaces.