Math 2.0

Differential geometry Definition

Shape operator

Builds on Regular surfaceSecond fundamental formFirst fundamental formTangent plane and normal vector of a surfaceSelf-adjoint operator

Definition

Let S⊆ℝ3S \subseteq \R^3 be a regular surface, let W⊆SW \subseteq S be open in SS, and let N ⁣:W→ℝ3N \colon W \to \R^3 be a smooth unit normal vector field. For p∈Wp \in W, the differential dNpdN_p maps the tangent plane TpST_p S into itself, because ⟨N,N⟩=1\ip{N}{N} = 1. The shape operator of SS at pp is the linear map Sp=−dNp ⁣:TpS→TpS.S_p = -dN_p \colon T_p S \to T_p S.

Remarks

The map SpS_p is self-adjoint for the inner product of TpST_p S, and ⟨Spv,w⟩=I ⁣Ip(v,w)\ip{S_p v}{w} = I\!I_p(v, w), where I ⁣IpI\!I_p is the polar form of the second fundamental form. In a local parametrisation, the matrix of SpS_p in the basis xu,xvx_u, x_v is (EFFG)−1(effg),\begin{pmatrix} E & F \\ F & G \end{pmatrix}^{-1} \begin{pmatrix} e & f \\ f & g \end{pmatrix}, with E,F,GE, F, G from the first fundamental form and e,f,ge, f, g from the second. Example: on the sphere of radius rr centred at 00 with the outer normal N(p)=p/rN(p) = p / r, one has dNp=1rid⁡dN_p = \frac{1}{r} \id, so Sp=−1rid⁡S_p = -\frac{1}{r} \id. With the inner normal, Sp=1rid⁡S_p = \frac{1}{r} \id. Non-example: on a Möbius strip there is no global continuous unit normal, so the shape operator is defined only locally, up to sign.

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFirst fundamental form, concept, no evidence levelFirst fundamental formconceptFunction, concept, no evidence levelFunctionconceptRegular surface, concept, no evidence levelRegular surfaceconceptSecond fundamental form, concept, no evidence levelSecond fundamental fo…conceptVector space, definition, no evidence levelVector spacedefinitionFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptInner product, definition, no evidence levelInner productdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptInner product space, definition, no evidence levelInner product spacedefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionAdjoint of a linear map, definition, no evidence levelAdjoint of a linear m…definitionGradient, definition, no evidence levelGradientdefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionSelf-adjoint operator, definition, no evidence levelSelf-adjoint operatordefinitionIterated derivative, definition, no evidence levelIterated derivativedefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementParametrised curve, definition, no evidence levelParametrised curvedefinitionParametrised surface, definition, no evidence levelParametrised surfacedefinitionTangent plane and normal vector of a surface, definition, no evidence levelTangent plane and nor…definitionShape operator, definition, no evidence levelShape operatordefinition
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.
  • no evidence level
  1. Euclidean space object
  2. First fundamental form concept
  3. Function concept
  4. Regular surface concept
  5. Second fundamental form concept
  6. Vector space definition
  7. Fréchet derivative definition
  8. Function of several variables concept
  9. Vector-valued function concept
  10. Inner product definition
  11. Linear map definition
  12. Limit of a function of several variables concept
  13. Partial derivative concept
  14. Inner product space definition
  15. Endomorphism definition
  16. Continuity in R^n definition
  17. Differentiable function of several variables definition
  18. Partial derivative definition
  19. Adjoint of a linear map definition
  20. Gradient definition
  21. Differentiable on an open set definition
  22. Higher partial derivatives definition
  23. Self-adjoint operator definition
  24. Iterated derivative definition
  25. C^k function characterised by continuous derivatives statement
  26. Parametrised curve definition
  27. Parametrised surface definition
  28. Tangent plane and normal vector of a surface definition
  29. Shape operator definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Differential_geometry_of_surfaces
About this page
Tag8RT3FGZPZY
URIm2://8RT3FGZPZY
Kinddefinition
WritingW0 stub
MSC53A05
DefinesShape operator
UsesRegular surface, Second fundamental form, First fundamental form, Tangent plane and normal vector of a surface, Self-adjoint operator
Changed2 November 2026

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