In thedifferential geometry of surfaces in three dimensions,umbilics orumbilical points are points on a surface that are locally spherical. At such points thenormal curvatures in all directions are equal, hence, bothprincipal curvatures are equal, and every tangent vector is aprincipal direction. The name "umbilic" comes from the Latinumbilicus (navel).

Umbilic points generally occur as isolated points in the elliptical region of the surface; that is, where theGaussian curvature is positive.

Thesphere is the only surface with non-zero curvature where every point is umbilic. A flat umbilic is an umbilic with zero Gaussian curvature. Themonkey saddle is an example of a surface with a flat umbilic and on theplane every point is a flat umbilic. A closed surface topologically equivalent to atorus may or may not have zero umbilics, but every closed surface of nonzeroEuler characteristic, embedded smoothly intoEuclidean space, has at least one umbilic.A famous conjecture ofConstantin Carathéodory dating from 1924 states that every smooth surface topologically equivalent to the sphere has at least two umbilics.^[1] The Conjecture was proven by Brendan Guilfoyle and Wilhelm Klingenberg and published in three parts^[2][3][4] concluding in 2024, the centenary of the Conjecture.

The three main types of umbilic points are elliptical umbilics, parabolic umbilics and hyperbolic umbilics. Elliptical umbilics have the threeridge lines passing through the umbilic and hyperbolic umbilics have just one. Parabolic umbilics are a transitional case with two ridges one of which is singular. Other configurations are possible for transitional cases. These cases correspond to theD₄⁻,D₅ andD₄⁺ elementary catastrophes of René Thom'scatastrophe theory.

Umbilics can also be characterised by the pattern of the principal directionvector field around the umbilic which typically form one of three configurations: star, lemon, and lemonstar (or monstar). Theindex 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 toDarboux and the names come from Hannay.^[5]

For surfaces withgenus 0 with isolated umbilics, e.g. an ellipsoid, the index of the principal direction vector field must be 2 by thePoincaré–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.^[6]

• configurations of lines of curvature near umbilics
• 
  Star
• 
  Monstar
• 
  Lemon

The classification of umbilics is closely linked to the classification of realcubic forms${\displaystyle a{x}^3+3b{x}^{2}y+3cx{y}^2+d{y}^3}$. A cubic form will have a number of root lines${\displaystyle \lambda (x,y)}$ such that the cubic form is zero for all real${\displaystyle \lambda }$. There are a number of possibilities including:

• Three distinct lines: anelliptical cubic form, standard model${\displaystyle x^{2}y-y^3}$.
• Three lines, two of which are coincident: aparabolic cubic form, standard model${\displaystyle x^{2}y}$.
• A single real line: ahyperbolic cubic form, standard model${\displaystyle x^{2}y+y^3}$.
• Three coincident lines, standard model${\displaystyle x^3}$.^[7]

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 theumbilic bracelet byChristopher Zeeman.^[7] 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${\displaystyle z^3+3\overline{\beta }{z}^2\overline{z}+3\beta z{\overline{z}}^2+{\overline{z}}^3}$ with a single complex parameter${\displaystyle \beta }$. Parabolic forms occur when${\displaystyle \beta =\frac{1}{3}(2e^{i\theta }+e^{-2i\theta })}$, the inner deltoid, elliptical forms are inside the deltoid and hyperbolic one outside. If${\displaystyle \left|\beta \right|=1}$ and${\displaystyle \beta }$ is not a cube root of unity then the cubic form is aright-angled cubic form which play a special role for umbilics. If${\displaystyle \left|\beta \right|=\frac{1}{3}}$ then two of the root lines are orthogonal.^[8]

A second cubic form, theJacobian is formed by taking theJacobian determinant of the vector valued function${\displaystyle F:{\mathbb{R}}^2\to {\mathbb{R}}^2}$,${\displaystyle F(x,y)=(x^2+y^2,a{x}^3+3b{x}^{2}y+3cx{y}^2+d{y}^3)}$. Up to a constant multiple this is the cubic form${\displaystyle b{x}^3+(2c-a)x^{2}y+(d-2b)x{y}^2-c{y}^3}$. Using complex numbers the Jacobian is a parabolic cubic form when${\displaystyle \beta =-2e^{i\theta }-e^{-2i\theta }}$, the outer deltoid in the classification diagram.^[8]

Any surface with an isolated umbilic point at the origin can be expressed as aMonge form parameterisation${\displaystyle z=\frac{1}{2}\kappa (x^2+y^2)+\frac{1}{3}(a{x}^3+3b{x}^{2}y+3cx{y}^2+d{y}^3)+\dots }$, where${\displaystyle \kappa }$ is the unique principal curvature. The type of umbilic is classified by the cubic form from the cubic part and corresponding Jacobian cubic form. 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.^[8]

The classification of umbilic points is as follows:^[8]

• Inside inner deltoid - elliptical umbilics
  • On inner circle - two ridge lines tangent
• On inner deltoid - parabolic umbilics
• Outside inner deltoid - hyperbolic umbilics
  • Inside outer circle - star pattern
  • On outer circle - birth of umbilics
  • Between outer circle and outer deltoid - monstar pattern
  • Outside outer deltoid - lemon pattern
• Cusps of the inner deltoid - cubic (symbolic) umbilics
• On the diagonals and the horizontal line - symmetrical umbilics with mirror symmetry

In a generic family of surfaces umbilics can be created, or destroyed, in pairs: thebirth of umbilics transition. Both umbilics will be hyperbolic, one with a star pattern and one with a monstar pattern. The outer circle in the diagram, a right angle cubic form, gives these transitional cases. Symbolic umbilics are a special case of this.^[8]

The elliptical umbilics and hyperbolic umbilics have distinctly differentfocal surfaces. A ridge on the surface corresponds to acuspidal 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.^[8]

A pointp in aRiemannian submanifold is umbilical if, atp, the (vector-valued)Second fundamental form is some normal vector tensor the induced metric (First fundamental form). Equivalently, for all vectorsU, V atp, II(U, V) = g_p(U, V)${\displaystyle \nu }$, where${\displaystyle \nu }$ is the mean curvature vector at p.

A submanifold is said to be umbilic (or all-umbilic) if this condition holds at every pointp. 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.

• umbilical – an anatomical term meaningof, or relating to the navel

• Darboux, Gaston (1896) [1887],Leçons sur la théorie génerale des surfaces: Volumes I–IV, Gauthier-Villars
  • Volume I
  • Volume II
  • Volume III
  • Volume IV
• Pictures of star, lemon, monstar, and further references

• Differential geometry of surfaces
• Surfaces