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| In [[Riemannian geometry]], the '''Gauss–Codazzi–Mainardi equations''' are fundamental equations in the theory of embedded [[hypersurface]]s in a [[Euclidean space]], and more generally [[submanifold]]s of [[Riemannian manifold]]s. They also have applications for embedded hypersurfaces of [[pseudo-Riemannian manifold]]s.
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| In the classical [[differential geometry of surfaces]], the Gauss–Codazzi–Mainardi equations consist of a pair of related equations. The first equation, sometimes called the '''Gauss equation''', relates the ''intrinsic curvature'' (or [[Gauss curvature]]) of the surface to the derivatives of the [[Gauss map]], via the [[second fundamental form]]. This equation is the basis for Gauss's [[theorema egregium]].<ref>{{harvnb|Gauss|1828}}.</ref> The second equation, sometimes called the '''Codazzi–Mainardi equation''', is a structural condition on the second derivatives of the Gauss map.
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| It was named for [[Gaspare Mainardi]] (1856) and [[Delfino Codazzi]] (1868–1869), who independently derived the result,<ref>{{harv|Kline|1972|p=885}}.</ref> although it was discovered earlier by {{harvtxt|Peterson|1853}}.<ref>{{harvnb|Ivanov|2001}}.</ref>
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| It incorporates the ''extrinsic curvature'' (or [[mean curvature]]) of the surface. The equations show that the components of the second fundamental form and its derivatives along the surface completely classify the surface up to a [[Euclidean transformation]], a theorem of [[Ossian Bonnet]].<ref>{{harvnb|Bonnet|1867}}.</ref>
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| ==Formal statement==
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| Let i : ''M'' ⊂ ''P'' be an ''n''-dimensional embedded submanifold of a Riemannian manifold ''P'' of dimension ''n''+''p''. There is a natural inclusion of the [[tangent bundle]] of ''M'' into that of ''P'' by the [[pushforward (differential)|pushforward]], and the [[cokernel]] is the [[normal bundle]] of ''M'':
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| :<math>0\rightarrow T_xM \rightarrow T_xP|_M \rightarrow T_x^\perp M\rightarrow 0.</math>
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| The metric splits this [[short exact sequence]], and so
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| :<math>TP|_M = TM\oplus T^\perp M.</math>
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| Relative to this splitting, the [[Levi-Civita connection]] ∇′ of ''P'' decomposes into tangential and normal components. For each ''X'' ∈ T''M'' and vector field ''Y'' on ''M'',
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| :<math>\nabla'_X Y = \top(\nabla'_X Y) + \bot(\nabla'_X Y).</math>
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| Let
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| :<math>\nabla_X Y = \top(\nabla'_X Y),\quad \alpha(X,Y) = \bot(\nabla'_X Y).</math>
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| '''Gauss' formula'''<ref>Terminology from Spivak, Volume III.</ref> now asserts that ∇<sub>X</sub> is the Levi-Civita connection for ''M'', and α is a ''symmetric'' [[vector-valued form]] with values in the normal bundle. It is often referred to as the [[second fundamental form]].
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| An immediate corollary is the '''Gauss equation'''. For ''X'', ''Y'', ''Z'', ''W'' ∈ T''M'',
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| :<math>\langle R'(X,Y)Z, W\rangle = \langle R(X,Y)Z, W\rangle + \langle \alpha(X,Z), \alpha(Y,W)\rangle -\langle \alpha(Y,Z), \alpha(X,W)\rangle </math>
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| where '''R'''′ is the [[Riemann curvature tensor]] of ''P'' and ''R'' is that of ''M''.
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| The '''Weingarten equation''' is an analog of the Gauss formula for a connection in the normal bundle. Let ''X'' ∈ T''M'' and ξ a normal vector field. Then decompose the ambient covariant derivative of ξ along ''X'' into tangential and normal components:
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| :<math>\nabla'_X\xi=\top (\nabla'_X\xi) + \bot(\nabla'_X\xi) = -A_\xi(X) + D_X(\xi).</math>
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| Then
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| # ''Weingarten's equation'': <math>\langle A_\xi X, Y\rangle = \langle \alpha(X,Y), \xi\rangle</math>
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| # ''D''<sub>X</sub> is a [[metric connection]] in the normal bundle.
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| There are thus a pair of connections: ∇, defined on the tangent bundle of ''M''; and ''D'', defined on the normal bundle of ''M''. These combine to form a connection on any tensor product of copies of T''M'' and T<sup>⊥</sup>''M''. In particular, they defined the covariant derivative of α:
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| :<math>(\tilde{\nabla}_X \alpha)(Y,Z) = D_X\left(\alpha(Y,Z)\right) - \alpha(\nabla_X Y,Z) - \alpha(Y,\nabla_X Z).</math>
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| The '''Codazzi–Mainardi equation''' is
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| :<math>\bot\left(R'(X,Y)Z\right) = (\tilde{\nabla}_X\alpha)(Y,Z) - (\tilde{\nabla}_Y\alpha)(X,Z).</math>
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| Since every [[immersion (mathematics)|immersion]] is, in particular, a local embedding, the above formulas also hold for immersions.
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| ==Gauss–Codazzi equations in classical differential geometry==
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| ===Statement of classical equations===
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| In classical [[differential geometry]] of surfaces, the Codazzi–Mainardi equations are expressed via the [[second fundamental form]] (''L'', ''M'', ''N''):
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| :<math>L_v-M_u=L\Gamma^1{}_{12} + M(\Gamma^2{}_{12}-\Gamma^1{}_{11}) - N\Gamma^2{}_{11}</math>
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| :<math>M_v-N_u=L\Gamma^1{}_{22} + M(\Gamma^2{}_{22}-\Gamma^1{}_{12}) - N\Gamma^2{}_{12}</math>
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| ===Derivation of classical equations===
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| Consider a [[parametric surface]] in Euclidean space,
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| :<math>\mathbf{r}(u,v) = (x(u,v),y(u,v),z(u,v))</math>
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| where the three component functions depend smoothly on ordered pairs (''u'',''v'') in some open domain ''U'' in the ''uv''-plane. Assume that this surface is '''regular''', meaning that the vectors '''r'''<sub>''u''</sub> and '''r'''<sub>''v''</sub> are [[linearly independent]]. Complete this to a [[basis of a vector space|basis]]{'''r'''<sub>u</sub>,'''r'''<sub>v</sub>,'''n'''}, by selecting a unit vector '''n''' normal to the surface. It is possible to express the second partial derivatives of '''r''' using the [[Christoffel symbols]] and the second fundamental form.
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| :<math>\bold{r}_{uu}=\Gamma^1{}_{11} \bold{r}_u + \Gamma^2{}_{11} \bold{r}_v + L \bold{n}</math>
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| :<math>\bold{r}_{uv}=\Gamma^1{}_{12} \bold{r}_u + \Gamma^2{}_{12} \bold{r}_v + M \bold{n}</math>
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| :<math>\bold{r}_{vv}=\Gamma^1{}_{22} \bold{r}_u + \Gamma^2{}_{22} \bold{r}_v + N \bold{n}</math>
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| [[Symmetry_of_second_derivatives#Clairaut.27s_theorem|Clairaut's theorem]] states that partial derivatives commute:
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| :<math>\left(\bold{r}_{uu}\right)_v=\left(\bold{r}_{uv}\right)_u</math>
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| If we differentiate '''r'''<sub>uu</sub> with respect to ''v'' and '''r'''<sub>uv</sub> with respect to ''u'', we get:
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| :<math>\left(\Gamma^1{}_{11}\right)_v \bold{r}_u + \Gamma^1{}_{11} \bold{r}_{uv} + \left(\Gamma^2{}_{11}\right)_v \bold{r}_v + \Gamma^2{}_{11} \bold{r}_{vv} + L_v \bold{n} + L \bold{n}_v </math> <math> = \left(\Gamma^1{}_{12}\right)_u \bold{r}_u + \Gamma^1{}_{12} \bold{r}_{uu} + \left(\Gamma_{12}^2\right)_u \bold{r}_v + \Gamma^2{}_{12} \bold{r}_{uv} + M_u \bold{n} + M \bold{n}_u</math>
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| Now substitute the above expressions for the second derivatives and equate the coefficients of '''n''':
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| :<math> M \Gamma^1{}_{11} + N \Gamma^2{}_{11} + L_v = L \Gamma^1{}_{12} + M \Gamma^2{}_{12} + M_u </math>
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| Rearranging this equation gives the first Codazzi–Mainardi equation.
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| The second equation may be derived similarly.
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| ==Mean curvature==
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| Let ''M'' be a smooth ''m''-dimensional manifold immersed in the (''m'' + ''k'')-dimensional smooth manifold ''P''. Let <math>e_1,e_2,\ldots, e_k</math> be a local orthonormal frame of vector fields normal to ''M''. Then we can write,
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| :<math>\alpha(X,Y) = \sum_{j=1}^k\alpha_j(X,Y)e_j</math>
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| If, now, <math>E_1,E_2,\ldots,E_m</math> is a local orthonormal frame (of tangent vector fields) on the same open subset of ''M'', then we can define the [[mean curvature]]s of the immersion by
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| :<math>H_j=\sum_{i=1}^m\alpha_j(E_i,E_i)</math>
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| In particular, if ''M'' is a hypersurface of ''P'', i.e. <math>k=1</math>, then there is only one mean curvature to speak of. The immersion is called [[minimal surface|minimal]] if all the <math>H_j</math> are identically zero.
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| Observe that the mean curvature is a trace, or average, of the second fundamental form, for any given component. Sometimes mean curvature is defined by multiplying the sum on the right-hand side by <math>1/m</math>.
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| We can now write the Gauss–Codazzi equations as
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| :<math>\langle R'(X,Y)Z, W\rangle = \langle R(X,Y)Z, W \rangle + \sum_{j=1}^k \alpha_j(X,Z) \alpha_j(Y,W) - \alpha_j(Y,Z) \alpha_j(X,W) </math>
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| Contracting the <math>Y,Z</math> components gives us
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| :<math> Ric'(X, W) = Ric(X,W) + \sum_{j=1}^k \langle R'(X,e_j)e_j,W\rangle+ \left(\sum_{i=1}^m\alpha_j(X,E_i) \alpha_j(E_i,W)\right) - H_j \alpha_j(X,W) </math> | |
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| Observe that the tensor in parentheses is symmetric and nonnegative-definite in <math>X,W</math>. Assuming that ''M'' is a hypersurface, this simplifies to
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| :<math> Ric'(X, W) = Ric(X,W) + \langle R'(X,n)n,W\rangle+ \left(\sum_{i=1}^mh(X,E_i) h(E_i,W)\right) - H h(X,W) </math>
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| where <math>n = e_1</math> and <math>h = \alpha_1</math> and <math>H = H_1</math>. In that case, one more contraction yields,
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| :<math> R' = R + 2 Ric'(n,n)+ \|h\|^2 - H^2 </math>
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| where <math>R'</math> and <math>R</math> are the respective scalar curvatures, and
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| :<math>\|h\|^2 = \sum_{i,j=1}^m h(E_i,E_j)^2</math>
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| If <math>k>1</math>, the scalar curvature equation might be more complicated.
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| We can already use these equations to draw some conclusions. For example, any minimal immersion<ref>{{harvnb|Takahashi|1966}}</ref> into the round sphere <math> x_1^2 + x_2^2 + \cdots + x_{m+k+1}^2 = 1 </math> must be of the form
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| :<math>\triangle x_j + \lambda x_j = 0</math>
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| where <math>j</math> runs from 1 to <math>m+k+1</math> and
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| :<math>\triangle = \sum_{i=1}^m \nabla_{E_i}\nabla_{E_i}</math>
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| is the [[Laplace–Beltrami operator|Laplacian]] on ''M'', and <math>\lambda>0</math> is a positive constant.
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| ==See also==
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| * [[Darboux frame]]
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| ==Notes==
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| {{reflist|29em}}
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| ==References==
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| * {{citation|first = Ossian|last = Bonnet|title=Memoire sur la theorie des surfaces applicables sur une surface donnee|journal=[[Journal de l'École Polytechnique]]|volume=25|pages=31–151|year=1867}}
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| * {{citation|title = Riemannian Geometry|first=Manfredo Perdigao | last = do Carmo | year = 1994 | others = Francis Flaherty}}
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| * {{citation|first=Delfino|last=Codazzi|year=1868–1869|title=Sulle coordinate curvilinee d'una superficie dello spazio|journal = Ann. Math. Pura applicata|volume=2|pages=101–19}}
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| * {{citation|first = Carl Friedrich|last = Gauss|title=Disquisitiones Generales circa Superficies Curvas|journal = Comm. Soc. Gott.|volume = 6|year = 1828|language=Latin|trans_title=General Discussions about Curved Surfaces}} ("General Discussions about Curved Surfaces")
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| * {{springer|id=P/p072450|title=Peterson–Codazzi equations|first=A.B.|last=Ivanov|year=2001}}
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| * {{citation|first=Morris|last=Kline|title=Mathematical Thought from Ancient to Modern Times|publisher=Oxford University Press|year=1972|isbn=0-19-506137-3}}
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| * {{citation|first=Gaspare|last=Mainardi|year=1856|title = Su la teoria generale delle superficie|journal=Giornale dell' Istituto Lombardo|volume=9|pages= 385–404}}
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| *{{citation|last=Peterson|first=Karl Mikhailovich|title=Über die Biegung der Flächen|publisher=Doctoral thesis, Dorpat University|year=1853}}.
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| *{{citation|first=Tsunero|last=Takahashi|title=Minimal Immersions of Riemannian Manifolds|journal=Journal of the Mathematical Society of Japan|year=1966}}
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| ==External links==
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| *[http://mathworld.wolfram.com/Peterson-Mainardi-CodazziEquations.html Peterson–Mainardi–Codazzi Equations – from Wolfram MathWorld]
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| *[http://eom.springer.de/p/p072450.htm Peterson–Codazzi Equations]
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| {{curvature}}
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| {{DEFAULTSORT:Gauss-Codazzi equations}}
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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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| [[Category:Surfaces]]
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