Hypersurfaces in space forms satisfying some curvature conditions
In [\textit{B. E. Abdalla} and \textit{F. Dillen}, Proc. Am. Math. Soc. 130, No. 6, 1805--1808 (2002; Zbl 1008.53021)] an example of a non-semisymmetric Ricci-symmetric quasi-Einstein austere hypersurface \(M^n\) isometrically immersed in a Euclidean space \(\mathbb R^{n+1}\), \(n>4\), was constructed. In the paper the authors state that, at every point of the hypersurface \(M^n\), the following condition \((\ast)\) is satisfied: \((n-2)(R\cdot C-C\cdot R)=Q(S,C)\). Here \(R\) is the Riemann curvature tensor and \(C\) is the Weyl conformal curvature tensor. Due to [\textit{R. Deszcz} et al., Kragujevac J. Math. 35, No. 2, 223--247 (2011; Zbl 1289.53037)], given a symmetric \((0, 2)\)-tensor \(A\) and a \((0, k)\)-tensor field \(B\), the Tachibana tensor \(Q(A,B)\) of \(A\) and \(B\) is a \((0, k+2)\)-tensor \(Q(A,B)(X_1,\dots, X_k,X,Y)= -T((X\wedge_A Y)X_1,\dots, X_k)-\dots -T(X_1,\dots, (X\wedge_A Y)X_k)\), where \((X\wedge_A Y)Z=A(Y,Z)X-A(X,Z)Y\). It is shown that non-conformally flat and non-Einstein hypersurfaces with vanishing scalar curvature having at every point two distinct principal curvatures, as well as some hypersurfaces having at every point three distinct principal curvatures, satisfy \((\ast)\), and certain examples of hypersurfaces are presented.
scientific article; zbMATH DE number 6522228
scientific article