Abstract. This short expository note deals with local rigidity problems for hypersurfaces of type number two in {\petit R}$^4$. The main theorem proves the existence of hypersurfaces with exactly one nontrivial isometric deformation. Such hypersurfaces were considered by E.~Cartan in 1916, but their actual existence has not been proved until recently.
AMSclassification. 53C20, 53C35, 53C40
Keywords. Riemannian manifold, semi-symmetric space, hypersurfaces in Euclidean spaces, isometric deformation