On the nullity of conformal Killing graphs in foliated Riemannian spaces

We deal with entire conformal Killing graphs, that is, graphs constructed through the flow generated by a complete conformal Killing vector field V on a Riemannian space , and which are defined over an integral leaf of the foliation orthogonal to V. Under a suitable restriction on the norm of the gradient of the function z which determines such a graph I (z) pound, we establish sufficient conditions to ensure that I (z) pound is totally geodesic. Afterwards, when the ambient space has constant sectional curvature, we obtain lower estimates for the index of minimum relative nullity of I (z) pound.