Certain conditions on the Ricci tensor of real hypersurfaces in quaternionic projective spaces
Let \(M\) be a connected hypersurface in a quaternionic projective space \(QP^m, m\geq 3\), endowed with a metric \(g\) of constant quaternionic sectional curvature 4. Let \(N\) be a unit local normal vector field on \(M\) and \(U_i=-J_iN\), \(i=1,2,3\) be a local basis of the quaternionic structure of \(QP^m.\) One defines a distribution \({\mathcal D}\) by \[ {\mathcal D}(x)=\{X \in T_xM \mid X \perp U_i(x), i=1, 2, 3\}, \] \(\forall x \in M,\) which is orthogonal to the structure vector fields \(\{U_1, U_2, U_3\}\) and invariant with respect to the structure tensors \(\{\Phi_1, \Phi_2, \Phi_3\},\) where \(\Phi_iX\) denotes the tangential componenet of \(J_iX, i=1,2,3,\forall X\) tangent to \(M.\) The first author studied and classified in [Int. J. Math. Math. Sci. 19, 193-197 (1996; Zbl 0838.53023)] real hypersurfaces such that the Ricci tensor satisfies \[ (\nabla_XS)(Y)=c\sum_{i=1}^3\{g(\Phi_iX,Y)U_i+f_i(Y)\Phi_iX\}, \tag{1} \] \(\forall X, Y \) tangent to \(M, c\) being a constant, where \(\nabla \) is the covariant derivative on \(M\) and \(f_i(X)=g(X,U_i).\) Real hypersurfaces of \(QP^m, m\geq 3,\) such that \(\Phi_iS=S\Phi_i, i=1,2,3,\) were classified, too. The equation (1) restricted to vector fields on \({\mathcal D}\) gives \[ (\nabla_XS)(Y)=c\sum_{i=1}^3g(\Phi_iX,Y)U_i, \tag{1'} \] \(\forall X, Y \in {\mathcal D}, c\) being a nonzero constant. The purpose of the present paper is to classify real hypersurfaces of \(QP^m\) such that (1') is satisfied and moreover \[ (S\Phi_i-\Phi_iS)=\theta_i(X)U_i \quad (i=1,2,3), \tag{2} \] \(\forall X \in {\mathcal D}\), \(\theta_i\) being a 1-form on \(M\). The following theorem is proved: Let \(M\) be a real hypersurface of \(OP^m,\) satisfying (1') and (2). Then \(M\) is congruent to an open subset of either a geodesic hypersphere or a tube of radius \(r\), \(0<r< {\pi \over 2},\) over a totally geodesic \(QP^k\), \(k=\overline {1, m-2},\) such that \(\cot^2r={2k+1\over 2m-2k-1}.\)
scientific article; zbMATH DE number 1759082