Ortega, Miguel; de Dios Pérez, Juan Some conditions on the Weingarten endomorphism of real hypersurfaces in quaternionic space forms. (English) Zbl 1022.53048 Tsukuba J. Math. 25, No. 2, 299-309 (2001). Let \(QM^m(c)\), \(m\geq 3\), \(c\neq 0\), be a quaternionic space form of curvature \(c\). For a connected real hypersurface \(M\) in it let \(g\) be the metric, \(A\) the Weingarten endomorphism, \(R\) the curvature operator, and \(D\) the maximal quaternionic distribution on \(M\). It is known that there do not exist hypersurfaces \(M\) in \(QM^m(c)\) with parallel \(A\), i.e., with \(\nabla A=0\). A weaker condition \(R\cdot A=0\) characterizes the so-called semiparallel hypersurfaces [see e.g. Ü. Lumiste, Submanifolds with parallel fundamental form, Handbook of differential geometry, Vol. I, 779-864 (2000; Zbl 0964.53002)]. In the present paper the symmetric tensor \(h\) is introduced on \(D\) by \(h(X,Y)= g(AX,Y)\) for any \(X,Y\in D\). It is proved that if there exists a smooth function \(\lambda:M\to R\) such that \(h=\lambda g\) then \(\lambda\) is constant. A classification is given for all such \(M\) without boundary in \(Q M^m (c)\). Furthermore, the following two conditions are considered: \[ g\bigl(R (AX,Y)Z-AR(X,Y) Z,W\bigr)=0, \]\[ g\biggl(\bigl(R (X,Y)A\bigr)Z+ \bigl(R(Z,X)A \bigr)Y+ \bigl(R(Y,Z)A \bigr)X,W \biggr)=0 \quad\text{for any}\quad X,Y,Z,W\in D; \] The second one is weaker than \(R\cdot A=0\). Also all hypersurfaces \(M\) without boundary satisfying one of these conditions are classified, using the results by J. Berndt [J. Reine Angew. Math. 419, 9-26 (1991; Zbl 0718.53017)]. Reviewer: Ülo Lumiste (Tartu) Cited in 1 Review MSC: 53C40 Global submanifolds 53C56 Other complex differential geometry Keywords:quaternionic space forms; relatively semiparallel hypersurfaces; Weingarten endomorphism; curvature operator; maximal quaternionic distribution Citations:Zbl 0964.53002; Zbl 0718.53017 PDFBibTeX XMLCite \textit{M. Ortega} and \textit{J. de Dios Pérez}, Tsukuba J. Math. 25, No. 2, 299--309 (2001; Zbl 1022.53048) Full Text: DOI