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  "body": "\n## Definition\n\nLet $S \\subseteq \\R^3$ be a [[regular surface|6080X3FAD4]], let $p \\in S$, and choose a smooth unit normal field near $p$.\nThe point $p$ is an *umbilical point* of $S$ if the two [[principal curvatures|FA8QE1SGP4]] of $S$ at $p$ are equal.\n\n## Remarks\n\nEquivalently, the [[shape operator|DQCD11X1VM]] is a multiple of the identity, $S_p = \\lambda \\id$.\nAnother 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]].\nThe condition does not depend on the choice of the normal.\nExample: on the sphere of radius $r$, $S_p = \\pm \\frac{1}{r} \\id$ at every point, so every point is umbilical; on a plane $S_p = 0$.\nNon-example: no point of a cylinder of radius $r$ is umbilical, since its principal curvatures are $1/r$ and $0$, up to sign.\nA connected regular surface all of whose points are umbilical is contained in a plane or in a sphere.\n",
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