Math 2.0

Differential geometry Definition

Mean curvature

Builds on Regular surfacePrincipal curvatureShape operatorFirst fundamental formSecond fundamental formTrace of a matrix

Definition

Let S⊆ℝ3S \subseteq \R^3 be a regular surface, let p∈Sp \in S, and choose a smooth unit normal field NN 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 mean curvature of SS at pp is H(p)=k1+k22=12tr⁡Sp.H(p) = \frac{k_1 + k_2}{2} = \frac{1}{2} \tr S_p.

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, H=eG−2fF+gE2(EG−F2).H = \frac{e G - 2 f F + g E}{2 (E G - F^2)}. Example: the cylinder of radius rr with the inner normal has k1=1/rk_1 = 1/r and k2=0k_2 = 0, so H=1/(2r)H = 1/(2r). The saddle z=xyz = xy has k1=1k_1 = 1 and k2=−1k_2 = -1 at the origin, so H=0H = 0 there. Non-example: replacing NN by −N-N changes HH into −H-H, so on a surface without a global unit normal, such as a Möbius strip, only ∣H∣\abs{H} is defined globally. Some authors define HH as k1+k2k_1 + k_2, without the factor 12\frac{1}{2}.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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First fundamental form, concept, no evidence levelFirst fundamental formconceptMatrix, 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 operatorconceptSquare matrix, definition, no evidence levelSquare matrixdefinitionTrace of a matrix, definition, no evidence levelTrace of a matrixdefinitionMean curvature, definition, no evidence levelMean 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. Matrix definition
  3. Principal curvature concept
  4. Regular surface concept
  5. Second fundamental form concept
  6. Shape operator concept
  7. Square matrix definition
  8. Trace of a matrix definition
  9. Mean curvature definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Mean_curvature
About this page
TagT2HPG1780R
URIm2://T2HPG1780R
Kinddefinition
WritingW0 stub
MSC53A05, 53A10
DefinesMean curvature
UsesRegular surface, Principal curvature, Shape operator, First fundamental form, Second fundamental form, Trace of a matrix
Changed2 November 2026

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