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| In [[differential geometry]], the '''second fundamental form''' (or '''shape tensor''') is a [[quadratic form]] on the [[tangent plane]] of a [[differential geometry of surfaces|smooth surface]] in the three dimensional [[Euclidean space]], usually denoted by <math>\mathrm{I\!I}</math> (read "two"). Together with the [[first fundamental form]], it serves to define extrinsic invariants of the surface, its [[principal curvature]]s. More generally, such a quadratic form is defined for a smooth [[hypersurface]] in a [[Riemannian manifold]] and a smooth choice of the unit normal vector at each point.
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| == Surface in R<sup>3</sup> ==
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| ===Motivation===
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| The second fundamental form of a [[parametric surface]] ''S'' in '''R'''<sup>3</sup> was introduced and studied by [[Carl Friedrich Gauss|Gauss]]. First suppose that the surface is the graph of a twice continuously differentiable function, ''z'' = ''f''(''x'',''y''), and that the plane ''z'' = 0 is [[tangent]] to the surface at the origin. Then ''f'' and its [[partial derivative]]s with respect to ''x'' and ''y'' vanish at (0,0). Therefore, the [[Taylor expansion]] of ''f'' at (0,0) starts with quadratic terms:
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| : <math> z=L\frac{x^2}{2} + Mxy + N\frac{y^2}{2} +
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| \mathrm{\scriptstyle{{\ }higher{\ }order{\ }terms}},</math>
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| and the second fundamental form at the origin in the coordinates ''x'', ''y'' is the [[quadratic form]]
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| : <math> L \, \text{d}x^2 + 2M \, \text{d}x \, \text{d}y + N \, \text{d}y^2. \,</math> <!--- "\'" improves the display of the formula. Do not delete --->
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| For a smooth point ''P'' on ''S'', one can choose the coordinate system so that the coordinate ''z''-plane is tangent to ''S'' at ''P'' and define the second fundamental form in the same way.
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| ===Classical notation===
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| The second fundamental form of a general parametric surface is defined as follows. Let {{nowrap|1='''r''' = '''r'''(''u'',''v'')}} be a regular parametrization of a surface in '''R'''<sup>3</sup>, where '''r''' is a smooth [[vector valued function]] of two variables. It is common to denote the partial derivatives of '''r''' with respect to ''u'' and ''v'' by '''r'''<sub>u</sub> and '''r'''<sub>v</sub>. Regularity of the parametrization means that '''r'''<sub>u</sub> and '''r'''<sub>v</sub> are linearly independent for any (''u'',''v'') in the domain of '''r''', and hence span the tangent plane to ''S'' at each point. Equivalently, the [[cross product]] '''r'''<sub>u</sub> × '''r'''<sub>v</sub> is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors '''n''':
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| :<math>\mathbf{n} = \frac{\mathbf{r}_u\times\mathbf{r}_v}{|\mathbf{r}_u\times\mathbf{r}_v|}.</math>
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| The second fundamental form is usually written as
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| :<math>\mathrm{I\!I} = L\, \text{d}u^2 + 2M\, \text{d}u\, \text{d}v + N\, \text{d}v^2, \,</math> <!--- "\'" improves the display of the formula. Do not delete --->
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| its matrix in the basis {'''r'''<sub>u</sub>, '''r'''<sub>v</sub>} of the tangent plane is
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| :<math> \begin{bmatrix}
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| L&M\\
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| M&N
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| \end{bmatrix}. </math>
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| The coefficients ''L'', ''M'', ''N'' at a given point in the parametric ''uv''-plane are given by the projections of the second partial derivatives of '''r''' at that point onto the normal line to ''S'' and can be computed with the aid of the [[dot product]] as follows:
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| :<math>L = \mathbf{r}_{uu} \cdot \mathbf{n}, \quad
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| M = \mathbf{r}_{uv} \cdot \mathbf{n}, \quad
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| N = \mathbf{r}_{vv} \cdot \mathbf{n}. </math>
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| ===Physicist's notation===
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| The second fundamental form of a general parametric surface ''S'' is defined as follows: Let '''r'''='''r'''(''u''<sup>1</sup>,''u''<sup>2</sup>) be a regular parametrization of a surface in '''R'''<sup>3</sup>, where '''r''' is a smooth [[vector valued function]] of two variables. It is common to denote the partial derivatives of '''r''' with respect to ''u''<sup>α</sup> by '''r'''<sub>α</sub>, α = 1, 2. Regularity of the parametrization means that '''r'''<sub>1</sub> and '''r'''<sub>2</sub> are linearly independent for any (''u''<sup>1</sub>,''u''<sup>2</sup>) in the domain of '''r''', and hence span the tangent plane to ''S'' at each point. Equivalently, the [[cross product]] '''r'''<sub>1</sub> × '''r'''<sub>2</sub> is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors '''n''':
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| :<math>\mathbf{n} = \frac{\mathbf{r}_1\times\mathbf{r}_2}{|\mathbf{r}_1\times\mathbf{r}_2|}.</math>
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| The second fundamental form is usually written as
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| :<math>\mathrm{I\!I} = b_{\alpha \beta} \, \text{d}u^{\alpha} \, \text{d}u^{\beta}. \,</math> <!--- "\'" improves the display of the formula. Do not delete --->
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| The equation above uses the [[Einstein notation|Einstein Summation Convention]].
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| The coefficients ''b''<sub>αβ</sub> at a given point in the parametric (''u''<sup>1</sup>, ''u''<sup>2</sup>)-plane are given by the projections of the second partial derivatives of '''r''' at that point onto the normal line to ''S'' and can be computed in terms of the normal vector "n" as follows:
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| :<math>b_{\alpha \beta} = r_{\alpha \beta}^{\ \ \gamma} n_{\gamma}. </math>
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| == Hypersurface in a Riemannian manifold ==
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| In [[Euclidean space]], the second fundamental form is given by
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| :<math>\mathrm{I\!I}(v,w) = -\langle d\nu(v),w\rangle\nu</math>
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| where <math>\nu</math> is the [[Gauss map]], and <math>d\nu</math> the [[pushforward (differential)|differential]] of <math>\nu</math> regarded as a [[vector valued differential form]], and the brackets denote the [[metric tensor]] of Euclidean space.
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| More generally, on a Riemannian manifold, the second fundamental form is an equivalent way to describe the [[shape operator]] (denoted by <math>S</math>) of a hypersurface,
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| :<math>\mathrm I\!\mathrm I(v,w)=\langle S(v),w\rangle\nu= -\langle \nabla_v n,w\rangle\nu=\langle n,\nabla_v w\rangle\nu,</math>
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| where <math>\nabla_v w</math> denotes the [[covariant derivative]] of the ambient manifold and <math>n</math> a field of normal vectors on the hypersurface. (If the [[affine connection]] is [[torsion tensor|torsion-free]], then the second fundamental form is symmetric.)
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| The sign of the second fundamental form depends on the choice of direction of <math>n</math> (which is called a co-orientation of the hypersurface - for surfaces in Euclidean space, this is equivalently given by a choice of [[orientability|orientation]] of the surface).
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| === Generalization to arbitrary codimension ===
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| The second fundamental form can be generalized to arbitrary [[codimension]]. In that case it is a quadratic form on the tangent space with values in the [[normal bundle]] and it can be defined by
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| :<math>\mathrm{I\!I}(v,w)=(\nabla_v w)^\bot, </math> | |
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| where <math>(\nabla_v w)^\bot </math> denotes the orthogonal projection of [[covariant derivative]] <math>\nabla_v w </math> onto the normal bundle.
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| In [[Euclidean space]], the [[Riemann curvature tensor|curvature tensor]] of a [[submanifold]] can be described by the following formula:
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| :<math>\langle R(u,v)w,z\rangle =\langle \mathrm I\!\mathrm I(u,z),\mathrm I\!\mathrm I(v,w)\rangle-\langle \mathrm I\!\mathrm I(u,w),\mathrm I\!\mathrm I(v,z)\rangle.</math>
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| This is called the '''[[Gauss-Codazzi equation|Gauss equation]]''', as it may be viewed as a generalization of Gauss's [[Theorema Egregium]]. The [[eigenvalue]]s of the second fundamental form, represented in an [[orthonormal basis]], are the '''[[principal curvature]]s''' of the surface. A collection of orthonormal [[eigenvector]]s are called the '''principal directions'''.
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| For general Riemannian manifolds one has to add the curvature of ambient space; if <math>N</math> is a manifold embedded in a [[Riemannian manifold]] (<math>M,g</math>) then the curvature tensor <math>R_N </math> of <math>N</math> with induced metric can be expressed using the second fundamental form and <math>R_M </math>, the curvature tensor of <math>M</math>:
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| :<math>\langle R_N(u,v)w,z\rangle = \langle R_M(u,v)w,z\rangle+\langle \mathrm I\!\mathrm I(u,z),\mathrm I\!\mathrm I(v,w)\rangle-\langle \mathrm I\!\mathrm I(u,w),\mathrm I\!\mathrm I(v,z)\rangle.</math>
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| ==See also==
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| *[[First fundamental form]]
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| *[[Gaussian curvature]]
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| *[[Gauss–Codazzi equations]]
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| *[[Shape operator]]
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| ==References==
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| * {{cite book|first=Heinrich|last=Guggenheimer|title=Differential Geometry|year=1977|publisher=Dover|chapter=Chapter 10. Surfaces|isbn=0-486-63433-7}}
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| *{{cite book | author=Kobayashi, Shoshichi and Nomizu, Katsumi | title = Foundations of Differential Geometry, Vol. 2 | publisher=Wiley-Interscience | year=1996 |edition=New |isbn = 0-471-15732-5}}
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| * {{cite book|last=Spivak|first=Michael|title=A Comprehensive introduction to differential geometry (Volume 3)|year=1999|publisher=Publish or Perish|isbn=0-914098-72-1}}
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| ==External links==
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| * A PhD thesis about the geometry of the second fundamental form by Steven Verpoort: https://repository.libis.kuleuven.be/dspace/bitstream/1979/1779/2/hierrrissiedan!.pdf
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| {{curvature}}
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| [[Category:Differential geometry]]
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| [[Category:Differential geometry of surfaces]]
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| [[Category:Riemannian geometry]]
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| [[Category:Curvature (mathematics)]]
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