Umbilical point
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical.
At such points the normal curvatures in all directions are equal, hence, both principal curvatures are equal, and every tangent vector is a principal direction.
The name "umbilic" comes from the Latin umbilicus (navel).
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.
A closed surface topologically equivalent to a torus may or may not have zero umbilics, but every closed surface of nonzero Euler characteristic, embedded smoothly into Euclidean space, has at least one umbilic.
An unproven conjecture of Constantin Carathéodory states that every smooth surface topologically equivalent to the sphere has at least two umbilics.
Parabolic umbilics are a transitional case with two ridges one of which is singular.
Umbilics can also be characterised by the pattern of the principal direction vector field around the umbilic which typically form one of three configurations: star, lemon, and lemonstar (or monstar).
The index of the vector field is either −½ (star) or ½ (lemon, monstar).
Elliptical and parabolic umbilics always have the star pattern, whilst hyperbolic umbilics can be star, lemon, or monstar.
This classification was first due to Darboux and the names come from Hannay.
[2] For surfaces with genus 0 with isolated umbilics, e.g. an ellipsoid, the index of the principal direction vector field must be 2 by the Poincaré–Hopf theorem.
Generic genus 0 surfaces have at least four umbilics of index ½.
An ellipsoid of revolution has two non-generic umbilics each of which has index 1.
A cubic form will have a number of root lines
There are a number of possibilities including: The equivalence classes of such cubics under uniform scaling form a three-dimensional real projective space and the subset of parabolic forms define a surface – called the umbilic bracelet by Christopher Zeeman.
[4] Taking equivalence classes under rotation of the coordinate system removes one further parameter and a cubic forms can be represent by the complex cubic form
is not a cube root of unity then the cubic form is a right-angled cubic form which play a special role for umbilics.
[5] A second cubic form, the Jacobian is formed by taking the Jacobian determinant of the vector valued function
Using complex numbers the Jacobian is a parabolic cubic form when
[5] Any surface with an isolated umbilic point at the origin can be expressed as a Monge form parameterisation
Whilst principal directions are not uniquely defined at an umbilic the limits of the principal directions when following a ridge on the surface can be found and these correspond to the root-lines of the cubic form.
The pattern of lines of curvature is determined by the Jacobian.
The outer circle in the diagram, a right angle cubic form, gives these transitional cases.
A ridge on the surface corresponds to a cuspidal edges so each sheet of the elliptical focal surface will have three cuspidal edges which come together at the umbilic focus and then switch to the other sheet.
For a hyperbolic umbilic there is a single cuspidal edge which switch from one sheet to the other.
[5] A point p in a Riemannian submanifold is umbilical if, at p, the (vector-valued) Second fundamental form is some normal vector tensor the induced metric (First fundamental form).
is the mean curvature vector at p. A submanifold is said to be umbilic (or all-umbilic) if this condition holds at every point p. This is equivalent to saying that the submanifold can be made totally geodesic by an appropriate conformal change of the metric of the surrounding (“ambient”) manifold.
For example, a surface in Euclidean space is umbilic if and only if it is a piece of a sphere.
