A characterization of almost-Einstein real hypersurfaces of quaternionic projective space (Q1375783)

Let \(M\) be a real hypersurface of a quaternionic projective space \(QP^m \), \(m\geq 3\). Denote by \(g\) the Riemannian metric, by \(R\) the Riemannian curvature tensor, by \(S\) the Ricci tensor, and by \(A\) the shape operator of \(M\). Any canonical local basis of the quaternionic Kähler structure of \(QP^m\) induces vector fields \(U_1\), \(U_2\), \(U_3\) on \(M\) by applying it to a local normal vector field. \(M\) is called almost-Einstein if \(SX=aX +b \sum^3_{i=1} g(AX,U_i) U_i\) for some real numbers \(a,b\) and all tangent vectors \(X\) of \(M\). The author proves that \(M\) is almost Einstein if and only if \(R(X,Y)SZ+ R(Y,Z)SX+ R(Z,X)SY =0\) holds for all vector fields \(X,Y,Z\) tangent to the maximal quaternionic subbundle of the tangent bundle of \(M\).