---
tag: NGHWJGCXS9
kind: definition
name: Umbilical point
msc: [53A05]
defines: XYJ6T5CKR6
uses: [6080X3FAD4, FA8QE1SGP4, DQCD11X1VM, 9KHQR16P1B, MAJ3DH7MZS]
sources:
  - url: https://en.wikipedia.org/wiki/Umbilical_point
---

## Definition

Let $S \subseteq \R^3$ be a [[regular surface|6080X3FAD4]], let $p \in S$, and choose a smooth unit normal field near $p$.
The point $p$ is an *umbilical point* of $S$ if the two [[principal curvatures|FA8QE1SGP4]] of $S$ at $p$ are equal.

## Remarks

Equivalently, the [[shape operator|DQCD11X1VM]] is a multiple of the identity, $S_p = \lambda \id$.
Another equivalent form is $I\!I_p = \lambda I_p$, with $I\!I_p$ the [[second fundamental form|9KHQR16P1B]] and $I_p$ the [[first fundamental form|MAJ3DH7MZS]].
The condition does not depend on the choice of the normal.
Example: on the sphere of radius $r$, $S_p = \pm \frac{1}{r} \id$ at every point, so every point is umbilical; on a plane $S_p = 0$.
Non-example: no point of a cylinder of radius $r$ is umbilical, since its principal curvatures are $1/r$ and $0$, up to sign.
A connected regular surface all of whose points are umbilical is contained in a plane or in a sphere.