Real hypersurfaces of a complex hyperbolic space satisfying a pointwise nullity condition (Q1574258)

The author proves that horospheres, geodesic hyperspheres and tubes over \(\mathbb{C}H^{n-1}\) are, locally, the only real hypersurfaces of the complex hyperbolic space satisfying some pointwise curvature identities such as, e.g.: \(R(X,Y)\xi=k(g(Y,\xi)X-g(X,\xi)Y)\) for a function \(k\) and with \( \xi=JN\), \(N\) being the unit normal to the hypersurface.