Math 2.0

Differential geometry Definition

Gaussian curvature

Builds on Regular surfacePrincipal curvatureShape operatorFirst fundamental formSecond fundamental formDeterminant

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. Let SpS_p be the shape operator at pp and let k1,k2k_1, k_2 be the principal curvatures at pp. The Gaussian curvature of SS at pp is K(p)=det⁡Sp=k1k2.K(p) = \det S_p = k_1 k_2. It does not depend on the choice of the unit normal.

Remarks

In a local parametrisation, with E,F,GE, F, G the coefficients of the first fundamental form and e,f,ge, f, g those of the second fundamental form, K=eg−f2EG−F2.K = \frac{e g - f^2}{E G - F^2}. Example: on the sphere of radius rr, Sp=±1rid⁡S_p = \pm \frac{1}{r} \id, so K=1/r2K = 1/r^2. On a cylinder k1=1/rk_1 = 1/r and k2=0k_2 = 0, so K=0K = 0; on the saddle z=xyz = xy at the origin k1=1k_1 = 1 and k2=−1k_2 = -1, so K=−1K = -1. Non-example: on that saddle, the normal curvatures in the orthogonal directions (1,0,0)(1, 0, 0) and (0,1,0)(0, 1, 0) are both 00. Their product 00 is not K=−1K = -1, because these directions are not principal directions. For the graph z=h(x,y)z = h(x, y) of a smooth function, K=(hxxhyy−hxy2)/(1+hx2+hy2)2K = (h_{xx} h_{yy} - h_{xy}^2) / (1 + h_x^2 + h_y^2)^2.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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First fundamental form, concept, no evidence levelFirst fundamental formconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionPrincipal 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 operatorconceptVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionGaussian curvature, definition, no evidence levelGaussian curvaturedefinition
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. Identity matrix object
  3. Matrix definition
  4. Principal curvature concept
  5. Regular surface concept
  6. Second fundamental form concept
  7. Shape operator concept
  8. Vector space definition
  9. Square matrix definition
  10. Linear map definition
  11. Multilinear map definition
  12. Alternating multilinear map definition
  13. Determinant definition
  14. Gaussian curvature definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Gaussian_curvature
About this page
Tag88VFJSZ8X2
URIm2://88VFJSZ8X2
Kinddefinition
WritingW0 stub
MSC53A05
DefinesGaussian curvature
UsesRegular surface, Principal curvature, Shape operator, First fundamental form, Second fundamental form, Determinant
Changed2 November 2026

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