  A spacelike 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 spacetime. In particular, for
conformally flat spacetimes (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.
