On 2-quasi-umbilical pseudosymmetric hypersurfaces in the Euclidean space (Q2470860)

If the curvature tensor \(R\) of a Riemannian manifold \(M\) satisfies the condition, that \(R(X,Y)\cdot R\) and \((X\land Y)\cdot R\) are linearly dependent for arbitrary vector fields \(X, Y\), then \(M\) is called pseudosymmetric. Furthermore, a hypersurface \(M^n\) \((n \geq 4)\) in a Riemannian manifold is called 2-quasi-umbilical if the mulitplicities of the eigenvalues of the shape operator \(A\) are 1, 1 and \(n-2\), so in diagonal form it is \(A= \text{diag}(a,b,c,\dots,c)\). In the present paper the author characterizes 2-quasi-umbilical hypersurfaces in Euclidean space \(E^{n+1}(n \geq 4)\) which are pseudosymmetric by the fact that the shape operator is \(A= \text{diag}(a,a,c,\dots,c)\) and \(R(X,Y)\cdot R = ac(X\land Y)\cdot R\).