Umbilical point

In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points that are locally spherical. At such points both principal curvatures are equal, and every tangent vector is a principal direction.

Umbilic points generally occur as isolated points in the elliptical region of the surface; that is, where the Gaussian curvature is positive. The sphere is the only surface with non-zero curvature where every point is umbilic. The monkey saddle is an example of a surface which has an umbilic at a point where the Gaussian curvature is zero.

There is a complex classification of umbilic points with elliptical, hyperbolic and parabolic umbilics. The classification determines the number of ridge lines passing through the umbilic (either 1 or 3) and the index of the principal direction vector field around the umbilic, which is either +½ or −½. The lines of curvature through umbilic points will typically form one of three configurations: star, lemon, and lemonstar (or monstar). Other configurations are possible for transitional cases.

Definition in higher dimension in Riemannian manifolds

A point p in a Riemannian submanifold is umbilical if, at p, the (vector-valued) Second fundamental form is the normal vector with the First fundamental form as its coefficient. Namely, for any vector U, V at p, II(U, V) = gx(U, V)\nu, where \nu is the mean curvature vector at p.

We can compare the conditions "totally geodesic" and "umbilical".

See also

References