In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points which 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.
 Lemon |
 Monstar |
 Star |
Definition in higher dimension in Riemannian manifolds
A point p in a Riemannian manifold is umbilical if, at p, the (vector-valued) Second fundamental form is the normal vector with the First fundamental form as it coefficient. Namely, for any vector U,V at p, II(U,V)=gx(U,V)ν, where νis the normal vector at p.
See also
- umbilical - an anatomical term meaning of, or relating to the navel.
- Umbilical
- Carathéodory conjecture
References
- Darboux, Gaston (1887,1889,1896), Leçons sur la théorie génerale des surfaces: Volume I, Volume II, Volume III, Volume IV, Gauthier-Villars
- Porteous, I. R. (1994), Geometric Differentiation, Cambridge University Press, ISBN 0-521-39063-X
- Pictures of star, lemon, monstar, and further references
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