Math 2.0

Differential geometry Definition

Umbilical point

Builds on Regular surfacePrincipal curvatureShape operatorSecond fundamental formFirst fundamental form

Definition

Let S⊆ℝ3S \subseteq \R^3 be a regular surface, let p∈Sp \in S, and choose a smooth unit normal field near pp. The point pp is an umbilical point of SS if the two principal curvatures of SS at pp are equal.

Remarks

Equivalently, the shape operator is a multiple of the identity, Sp=λid⁡S_p = \lambda \id. Another equivalent form is I ⁣Ip=λIpI\!I_p = \lambda I_p, with I ⁣IpI\!I_p the second fundamental form and IpI_p the first fundamental form. The condition does not depend on the choice of the normal. Example: on the sphere of radius rr, Sp=±1rid⁡S_p = \pm \frac{1}{r} \id at every point, so every point is umbilical; on a plane Sp=0S_p = 0. Non-example: no point of a cylinder of radius rr is umbilical, since its principal curvatures are 1/r1/r and 00, up to sign. A connected regular surface all of whose points are umbilical is contained in a plane or in a sphere.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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First fundamental form, concept, no evidence levelFirst fundamental formconceptPrincipal curvature, concept, no evidence levelPrincipal curvatureconceptRegular surface, concept, no evidence levelRegular surfaceconceptSecond fundamental form, concept, no evidence levelSecond fundamental fo…conceptShape operator, concept, no evidence levelShape operatorconceptUmbilical point, definition, no evidence levelUmbilical pointdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. First fundamental form concept
  2. Principal curvature concept
  3. Regular surface concept
  4. Second fundamental form concept
  5. Shape operator concept
  6. Umbilical point definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Umbilical_point
About this page
TagNGHWJGCXS9
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Kinddefinition
WritingW0 stub
MSC53A05
DefinesUmbilical point
UsesRegular surface, Principal curvature, Shape operator, Second fundamental form, First fundamental form
Changed2 November 2026

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