View source for Gauss–Codazzi equations

{{Short description|Fundamental formulas linking the metric and curvature tensor of a manifold}}
In [[Riemannian geometry]] and [[pseudo-Riemannian manifold|pseudo-Riemannian geometry]], the '''Gauss–Codazzi equations''' (also called the '''Gauss–Codazzi–Weingarten-Mainardi equations''' or '''Gauss–Peterson–Codazzi formulas'''<ref>{{harvtxt|Toponogov|2006}}</ref>) are fundamental formulas that link together the induced metric and second fundamental form of a submanifold of (or immersion into) a [[Riemannian manifold|Riemannian]] or [[pseudo-Riemannian manifold]].

The equations were originally discovered in the context of surfaces in three-dimensional [[Euclidean space]]. In this context, the first equation, often called the '''Gauss equation''' (after its discoverer [[Carl Friedrich Gauss]]), says that the [[Gauss curvature]] of the surface, at any given point, is dictated by the derivatives of the Gauss map at that point, as encoded by the [[second fundamental form]].<ref>This equation is the basis for Gauss's [[theorema egregium]]. {{harvnb|Gauss|1828}}.</ref> The second equation, called the '''Codazzi equation''' or '''Codazzi-Mainardi equation''', states that the [[covariant derivative]] of the second fundamental form is fully symmetric. It is 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 [[Karl Mikhailovich Peterson]].<ref>{{harvtxt|Peterson|1853}}</ref><ref>{{harvnb|Ivanov|2001}}.</ref>

==Formal statement==
Let <math>i \colon M \subset P</math> be an ''n''-dimensional embedded submanifold of a Riemannian manifold ''P'' of dimension <math>n+p</math>.  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'':
:<math>0 \rightarrow T_xM \rightarrow T_xP|_M \rightarrow T_x^\perp M \rightarrow 0.</math>

The metric splits this [[short exact sequence]], and so
:<math>TP|_M = TM\oplus T^\perp M.</math>

Relative to this splitting, the [[Levi-Civita connection]] <math>\nabla'</math> of ''P'' decomposes into tangential and normal components.  For each <math>X\in TM</math> and vector field ''Y'' on ''M'',
:<math>\nabla'_X Y = \top\left(\nabla'_X Y\right) + \bot\left(\nabla'_X Y\right).</math>

Let
:<math>\nabla_X Y = \top\left(\nabla'_X Y\right),\quad \alpha(X, Y) = \bot\left(\nabla'_X Y\right).</math>

The '''Gauss formula'''<ref>Terminology from Spivak, Volume III.</ref> now asserts that <math>\nabla_X</math> is the [[Levi-Civita connection]] for ''M'', and <math>\alpha</math> is a ''symmetric'' [[vector-valued form]] with values in the normal bundle.  It is often referred to as the [[second fundamental form]].

An immediate corollary is the '''Gauss equation for the curvature tensor'''.  For <math>X, Y, Z, W \in TM</math>,
:<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>
where <math>R'</math> is the [[Riemann curvature tensor]] of ''P'' and ''R'' is that of ''M''.

The [[Weingarten equations|'''Weingarten equation''']] is an analog of the Gauss formula for a connection in the normal bundle.  Let <math>X \in TM</math> and <math>\xi</math> a normal vector field.  Then decompose the ambient covariant derivative of <math>\xi</math> along ''X'' into tangential and normal components:
:<math>\nabla'_X\xi = \top \left(\nabla'_X\xi\right) + \bot\left(\nabla'_X\xi\right) = -A_\xi(X) + D_X(\xi).</math>

Then
# ''Weingarten's equation'': <math>\langle A_\xi X, Y\rangle = \langle \alpha(X, Y), \xi\rangle</math>
# ''D''<sub>X</sub> is a [[metric connection]] in the normal bundle.

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 <math>\alpha</math>:
:<math>\left(\tilde{\nabla}_X \alpha\right)(Y, Z) = D_X\left(\alpha(Y, Z)\right) - \alpha\left(\nabla_X Y, Z\right) - \alpha\left(Y, \nabla_X Z\right).</math>

The '''Codazzi–Mainardi equation''' is
:<math>\bot\left(R'(X, Y)Z\right) = \left(\tilde{\nabla}_X\alpha\right)(Y, Z) - \left(\tilde{\nabla}_Y\alpha\right)(X, Z).</math>

Since every [[immersion (mathematics)|immersion]] is, in particular, a local embedding, the above formulas also hold for immersions.

==Gauss–Codazzi equations in classical differential geometry==

===Statement of classical equations===
In classical [[differential geometry]] of surfaces, the Codazzi–Mainardi equations are expressed via the [[second fundamental form]] (''L'', ''M'', ''N''):
:<math>L_v-M_u = L\Gamma^1{}_{12} + M\left({\Gamma^2}_{12} - {\Gamma^1}_{11}\right) - N{\Gamma^2}_{11}</math>
:<math>M_v-N_u = L\Gamma^1{}_{22} + M\left({\Gamma^2}_{22} - {\Gamma^1}_{12}\right) - N{\Gamma^2}_{12}</math>

The Gauss formula, depending on how one chooses to define the Gaussian curvature, may be a [[tautology (logic)|tautology]]. It can be stated as
:<math>K = \frac{LN - M^2}{eg - f^2},</math>
where (''e'', ''f'', ''g'') are the components of the first fundamental form.

===Derivation of classical equations===
Consider a [[parametric surface]] in Euclidean 3-space,

:<math>\mathbf{r}(u,v) = (x(u,v),y(u,v),z(u,v))</math>

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''' (vectors of <math>\mathbb{R^3}</math>) with the [[Christoffel symbols]] and the elements of the second fundamental form. We choose the first two components of the basis as they are intrinsic to the surface and intend to prove intrinsic property of the [[Gaussian curvature]]. The last term in the basis is extrinsic.
:<math>\mathbf{r}_{uu} = {\Gamma^1}_{11} \mathbf{r}_u + {\Gamma^2}_{11} \mathbf{r}_v + L \mathbf{n}</math>
:<math>\mathbf{r}_{uv} = {\Gamma^1}_{12} \mathbf{r}_u + {\Gamma^2}_{12} \mathbf{r}_v + M \mathbf{n}</math>
:<math>\mathbf{r}_{vv} = {\Gamma^1}_{22} \mathbf{r}_u + {\Gamma^2}_{22} \mathbf{r}_v + N \mathbf{n}</math>

[[Symmetry of second derivatives#Schwarz's theorem|Clairaut's theorem]] states that partial derivatives commute:
:<math>\left(\mathbf{r}_{uu}\right)_v = \left(\mathbf{r}_{uv}\right)_u</math>

If we differentiate '''r'''<sub>uu</sub> with respect to ''v'' and '''r'''<sub>uv</sub> with respect to ''u'', we get:

:<math>\left({\Gamma^1}_{11}\right)_v \mathbf{r}_u + {\Gamma^1}_{11} \mathbf{r}_{uv} + \left({\Gamma^2}_{11}\right)_v \mathbf{r}_v + {\Gamma^2}_{11} \mathbf{r}_{vv} + L_v \mathbf{n} + L \mathbf{n}_v </math><math> = \left({\Gamma^1}_{12}\right)_u \mathbf{r}_u + {\Gamma^1}_{12} \mathbf{r}_{uu} + \left(\Gamma_{12}^2\right)_u \mathbf{r}_v + {\Gamma^2}_{12} \mathbf{r}_{uv} + M_u \mathbf{n} + M \mathbf{n}_u</math>

Now substitute the above expressions for the second derivatives and equate the coefficients of '''n''':
:<math> M {\Gamma^1}_{11} + N {\Gamma^2}_{11} + L_v = L {\Gamma^1}_{12} + M {\Gamma^2}_{12} + M_u </math>

Rearranging this equation gives the first Codazzi–Mainardi equation.

The second equation may be derived similarly.

==Mean curvature==

Let ''M'' be a smooth ''m''-dimensional manifold immersed in the (''m''&nbsp;+&nbsp;''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,

:<math>\alpha(X, Y) = \sum_{j=1}^k\alpha_j(X, Y)e_j.</math>

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

:<math>H_j=\sum_{i=1}^m\alpha_j(E_i, E_i).</math>

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.

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>.

We can now write the Gauss–Codazzi equations as

:<math>\langle R'(X, Y)Z, W \rangle = \langle R(X,Y)Z, W \rangle + \sum_{j=1}^k \left(\alpha_j(X,Z) \alpha_j(Y, W) - \alpha_j(Y, Z) \alpha_j(X, W)\right). </math>

Contracting the <math>Y, Z</math> components gives us

:<math>\operatorname{Ric}'(X, W) = \operatorname{Ric}(X,W) + \sum_{j=1}^k \langle R'(X, e_j)e_j, W\rangle +  \sum_{j=1}^k  \left(\sum_{i=1}^m\alpha_j(X, E_i) \alpha_j(E_i, W)- H_j \alpha_j(X, W)\right).</math>

When ''M'' is a hypersurface, this simplifies to

:<math>\operatorname{Ric}'(X, W) = \operatorname{Ric}(X, W) + \langle R'(X, n)n, W \rangle + \sum_{i=1}^mh(X, E_i) h(E_i, W) - H h(X, W)</math>

where <math>n = e_1,</math> <math>h = \alpha_1</math> and <math>H = H_1</math>.  In that case, one more contraction yields,

:<math>R' = R + 2 \operatorname{Ric}'(n, n) + \|h\|^2 - H^2</math>

where <math>R'</math> and <math>R</math> are the scalar curvatures of ''P'' and ''M'' respectively, and

:<math>\|h\|^2 = \sum_{i,j=1}^m h(E_i, E_j)^2.</math>

If <math>k>1</math>, the scalar curvature equation might be more complicated.

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

:<math>\Delta x_j + \lambda x_j = 0</math>

where <math>j</math> runs from 1 to <math>m + k + 1</math> and

:<math>\Delta = \sum_{i=1}^m \nabla_{E_i}\nabla_{E_i}</math>

is the [[Laplace–Beltrami operator|Laplacian]] on ''M'', and <math>\lambda > 0</math> is a positive constant.

==See also==
* [[Darboux frame]]

==Notes==
{{reflist|29em}}

==References==
{{sfn whitelist |CITEREFIvanov2001}}
'''Historical references'''
* {{citation|first = Ossian|last = Bonnet|authorlink=Ossian 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}}
* {{citation|first=Delfino|last=Codazzi|authorlink=Delfino Codazzi| year=1868–1869|title=Sulle coordinate curvilinee d'una superficie dello spazio|journal = Ann. Mat. Pura Appl.|volume=2|pages=101–19|doi=10.1007/BF02419605 |s2cid=177803350 |url=https://zenodo.org/record/1756343 }}
* {{citation|first = Carl Friedrich|last = Gauss|authorlink = Carl Friedrich  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")
* {{springer|id=P/p072450|title=Peterson–Codazzi equations|first=A.B.|last=Ivanov|year=2001}}
* {{citation|first=Morris|last=Kline|authorlink=Morris Kline|title=Mathematical Thought from Ancient to Modern Times|publisher=[[Oxford University Press]]|year=1972|isbn=0-19-506137-3|url-access=registration|url=https://archive.org/details/mathematicalthou00morr}}
* {{citation|first=Gaspare|last=Mainardi|authorlink=Gaspare Mainardi| year=1856|title = Su la teoria generale delle superficie|journal=Giornale Dell' Istituto Lombardo|volume=9|pages= 385–404}}
*{{citation|last=Peterson|first=Karl Mikhailovich|authorlink=Karl Mikhailovich Peterson| title=Über die Biegung der Flächen|publisher=Doctoral thesis, Dorpat University|year=1853}}.
'''Textbooks'''
* do Carmo, Manfredo P. ''Differential geometry of curves & surfaces.'' Revised & updated second edition. Dover Publications, Inc., Mineola, NY, 2016. xvi+510 pp. {{ISBN|978-0-486-80699-0|0-486-80699-5}}
* do Carmo, Manfredo Perdigão. ''Riemannian geometry.'' Translated from the second Portuguese edition by Francis Flaherty. Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, 1992. xiv+300 pp. {{ISBN|0-8176-3490-8}}
* Kobayashi, Shoshichi; Nomizu, Katsumi. ''Foundations of differential geometry. Vol. II.'' Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney 1969 xv+470 pp.
* O'Neill, Barrett. ''Semi-Riemannian geometry. With applications to relativity.'' Pure and Applied Mathematics, 103. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. xiii+468 pp. {{ISBN|0-12-526740-1}}
* {{cite book |last1=Toponogov |first1=Victor Andreevich |title=Differential geometry of curves and surfaces: A concise guide |date=2006 |publisher=Birkhäuser |location=Boston |isbn=978-0-8176-4384-3}}
'''Articles'''
*{{citation|first=Tsunero|last=Takahashi|title=Minimal immersions of Riemannian manifolds|journal=Journal of the Mathematical Society of Japan|year=1966|volume=18 |issue=4 |doi=10.2969/jmsj/01840380 |s2cid=122849496 |doi-access=free}}
* Simons, James. ''Minimal varieties in riemannian manifolds.'' Ann. of Math. (2) 88 (1968), 62–105.
* [https://faculty.ksu.edu.sa/sites/default/files/lec_30.pdf]
* [https://www.cis.upenn.edu/~cis6100/gma-v2-chap20.pdf]
* [https://liavas.net/courses/math430/files/Surfaces_part3.pdf]

==External links==
*[http://mathworld.wolfram.com/Peterson-Mainardi-CodazziEquations.html Peterson–Mainardi–Codazzi Equations – from Wolfram MathWorld]
*[https://encyclopediaofmath.org/wiki/Peterson-Codazzi_equations Peterson–Codazzi Equations]

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