Math 2.0

Differential geometry Definition

Second fundamental form

Builds on Regular surfaceFirst fundamental formTangent plane and normal vector of a surfaceDot product

Definition

Let S⊆ℝ3S \subseteq \R^3 be a regular surface, let x ⁣:U→Sx \colon U \to S be a local parametrisation, and let N=xu×xv/∥xu×xv∥N = x_u \times x_v / \norm{x_u \times x_v} be the unit normal vector. The coefficients of the second fundamental form of SS with respect to xx are e=⟨xuu,N⟩,f=⟨xuv,N⟩,g=⟨xvv,N⟩.e = \ip{x_{uu}}{N}, \qquad f = \ip{x_{uv}}{N}, \qquad g = \ip{x_{vv}}{N}. At p=x(q)p = x(q), the second fundamental form is the quadratic form I ⁣IpI\!I_p on the tangent plane TpST_p S with I ⁣Ip(axu+bxv)=ea2+2fab+gb2I\!I_p(a x_u + b x_v) = e a^2 + 2 f ab + g b^2.

Remarks

One writes I ⁣I=e du2+2f du dv+g dv2I\!I = e \, du^2 + 2 f \, du \, dv + g \, dv^2. Equivalently, I ⁣Ip(w)=−⟨dNp(w),w⟩I\!I_p(w) = -\ip{dN_p(w)}{w}, where dNpdN_p is the differential of the unit normal field; this shows that I ⁣IpI\!I_p depends only on SS and the choice of NN. Replacing NN by −N-N changes the sign of I ⁣II\!I. Example: for the cylinder x(θ,z)=(rcos⁡θ,rsin⁡θ,z)x(\theta, z) = (r \cos \theta, r \sin \theta, z) with the inner normal N=−(cos⁡θ,sin⁡θ,0)N = -(\cos \theta, \sin \theta, 0), one gets e=re = r, f=0f = 0 and g=0g = 0. For the plane, xuux_{uu}, xuvx_{uv} and xvvx_{vv} vanish for a linear parametrisation, so I ⁣I=0I\!I = 0. Non-example: on a Möbius strip there is no continuous unit normal on the whole surface, so the second fundamental form is defined only locally, up to sign. For w∈TpSw \in T_p S with Ip(w)=1I_p(w) = 1, the number I ⁣Ip(w)I\!I_p(w) is the normal curvature of SS in the direction ww, where IpI_p is the first fundamental form.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

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Coordinate space, object, no evidence levelCoordinate spaceobjectEuclidean space, object, no evidence levelEuclidean spaceobjectFirst fundamental form, concept, no evidence levelFirst fundamental formconceptFunction, concept, no evidence levelFunctionconceptRegular surface, concept, no evidence levelRegular surfaceconceptDot product, definition, no evidence levelDot productdefinitionFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptContinuity 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 derivativedefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionGradient, definition, no evidence levelGradientdefinitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated 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…definitionSecond fundamental form, definition, no evidence levelSecond fundamental fo…definition
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. Coordinate space object
  2. Euclidean space object
  3. First fundamental form concept
  4. Function concept
  5. Regular surface concept
  6. Dot product definition
  7. Fréchet derivative definition
  8. Function of several variables concept
  9. Vector-valued function concept
  10. Limit of a function of several variables concept
  11. Partial derivative concept
  12. Continuity in R^n definition
  13. Differentiable function of several variables definition
  14. Partial derivative definition
  15. Differentiable on an open set definition
  16. Gradient definition
  17. Higher partial derivatives definition
  18. Iterated derivative definition
  19. C^k function characterised by continuous derivatives statement
  20. Parametrised curve definition
  21. Parametrised surface definition
  22. Tangent plane and normal vector of a surface definition
  23. Second fundamental form definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Second_fundamental_form
About this page
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WritingW0 stub
MSC53A05
DefinesSecond fundamental form
UsesRegular surface, First fundamental form, Tangent plane and normal vector of a surface, Dot product
Changed2 November 2026

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