Umbilical-Type Surfaces in SpaceTime

A space-like surface S immersed in a 4-dimensional Lorentzian manifold will be said to be umbilical along a direction N normal to S if the second fundamental form along N is proportional to the first fundamental form of S. In particular, S is pseudo-umbilical if it is umbilical along the mean curvature vector field H and (totally) umbilical if it is umbilical along all possible normal directions. The possibility that the surface be umbilical along the unique normal direction orthogonal to H—“ortho-umbilical” surface—is also considered. I prove that the necessary and sufficient condition for S to be umbilical along a normal direction is that two independent Weingarten operators (and, a fortiori, all of them) commute, or equivalently that the shape tensor be diagonalizable on S. The umbilical direction is then uniquely determined. This can be seen to be equivalent to a condition relating the normal curvature and the appropriate part of the Riemann tensor of the space time. In particular, for conformally flat space-times (including Lorentz space forms) the necessary and sufficient condition is that the normal curvature vanishes. Some further consequences are analyzed, and the extension of the main results to arbitrary signatures and higher dimensions is briefly discussed.