Some conditions on the Weingarten endomorphism of real hypersurfaces in quaternionic space forms (Q1848095)

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. \textit{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 \textit{J. Berndt} [J. Reine Angew. Math. 419, 9-26 (1991; Zbl 0718.53017)].