View source for Gaussian curvature

''of a surface''

The product of the principal curvatures (cf. [[Principal curvature|Principal curvature]]) of a regular surface at a given point.

If

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is the [[First fundamental form|first fundamental form]] of the surface and

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is the [[Second fundamental form|second fundamental form]] of the surface, then the Gaussian curvature can be computed by the formula

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The Gaussian curvature is identical with the Jacobi determinant of the [[Spherical map|spherical map]]:

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g0435905.png" /> is a point on the surface, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g0435906.png" /> is the area of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g0435907.png" /> which contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g0435908.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g0435909.png" /> is the area of the spherical image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359010.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359011.png" /> is the diameter of the domain. The Gaussian curvature is positive at an [[Elliptic point|elliptic point]], negative at a [[Hyperbolic point|hyperbolic point]], and is zero at a [[Parabolic point|parabolic point]] or a [[Flat point|flat point]]. It may be expressed in terms of the coefficients of the first fundamental form and their derivatives alone (the [[Gauss theorem|Gauss theorem]]), viz.

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where

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Since the Gaussian curvature depends on the metric only, i.e. on the coefficients of the first fundamental form, the Gaussian curvature is invariant under isometric deformation (cf. [[Deformation, isometric|Deformation, isometric]]). The Gaussian curvature plays a special role in the theory of surfaces, and many formulas are available for its computation, [[#References|[2]]].

The concept was introduced by C.F. Gauss [[#References|[1]]], and was named after him.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.F. Gauss,   "Allgemeine Flächentheorie" , W. Engelmann , Leipzig  (1900)  (Translated from Latin)  {{MR|}}  {{ZBL|31.0599.11}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Reichardt,   "Einführung in die Differentialgeometrie" , Springer  (1960)  {{MR|0116267}} {{ZBL|0091.34001}} </TD></TR></table>



====Comments====
The total Gaussian curvature (often also abbreviated to total curvature) is the quantity

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(See also [[Gauss–Bonnet theorem|Gauss–Bonnet theorem]].)

For a smooth space curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359015.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359016.png" />, the total curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359018.png" /> is defined as the length of the spherical image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359019.png" /> (cf. also [[Spherical indicatrix|Spherical indicatrix]]) and can be expressed using the [[Frénet formulas|Frénet formulas]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359022.png" /> for a Frénet frame (cf. [[Frénet trihedron|Frénet trihedron]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359023.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043590/g04359024.png" /> by

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