Umbilical point
Encyclopedia
In the differential geometry of surfaces
in three dimensions, umbilics or umbilical points are points that are locally spherical. At such points both principal curvature
s 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.
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)
, where 
is the mean curvature vector at p.
We can compare the conditions "totally geodesic" and "umbilical".
Differential geometry of surfaces
In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric....
in three dimensions, umbilics or umbilical points are points that are locally spherical. At such points both principal curvature
Principal curvature
In differential geometry, the two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point. They measure how the surface bends by different amounts in different directions at that point.-Discussion:...
s 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
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...
is positive. The sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
is the only surface with non-zero curvature where every point is umbilic. The monkey saddle
Monkey saddle
In mathematics, the monkey saddle is the surface defined by the equation z = x^3 - 3xy^2. \, It belongs to the class of saddle surfaces and its name derives from the observation that a saddle for a monkey requires three depressions: two for the legs, and one for the tail...
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
Index (mathematics)
The word index is used in variety of senses in mathematics.- General :* In perhaps the most frequent sense, an index is a number or other symbol that indicates the location of a variable in a list or array of numbers or other mathematical objects. This type of index is usually written as a...
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 submanifoldRiemannian submanifold
A Riemannian submanifold N of a Riemannian manifold M is a submanifold of M equipped with the Riemannian metric inherited from M. The image of an isometric immersion is a Riemannian submanifold....
is umbilical if, at p, the (vector-valued) Second fundamental form is the normal vector with the First fundamental form
First fundamental form
In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of R3. It permits the calculation of curvature and metric properties of a surface such as length and...
as its coefficient. Namely, for any vector U, V at p, II(U, V) = gx(U, V)
, where is the mean curvature vector at p.We can compare the conditions "totally geodesic" and "umbilical".
See also
- umbilical – an anatomical term meaning of, or relating to the navel
- Carathéodory conjectureCarathéodory conjectureThe Carathéodory conjecture is a mathematical conjecture attributed to Constantin Carathéodory by Hans Ludwig Hamburger in a session of the Berlin Mathematical Society in 1924, [1]. Other early references are the Invited Lecture [3] of Stefan Cohn-Vossen at the International Congress of...