Gauss–Codazzi equations

In Riemannian geometry, the Gauss–Codazzi–Mainardi equations are fundamental equations in the theory of embedded hypersurfaces in a Euclidean space, and more generally submanifolds of Riemannian manifolds. They also have applications for embedded hypersurfaces of pseudo-Riemannian manifolds.

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.[1] The second equation, sometimes called the Codazzi–Mainardi equation, is a structural condition on the second derivatives of the Gauss map. It was named for Gaspare Mainardi (1856) and Delfino Codazzi (1868–1869), who independently derived the result,[2] although it was discovered earlier by Peterson (1853).[3] 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.[4]

Formal statement

Let i : MP 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, and the cokernel is the normal bundle of M:

${\displaystyle 0\rightarrow T_{x}M\rightarrow T_{x}P|_{M}\rightarrow T_{x}^{\perp }M\rightarrow 0.}$

The metric splits this short exact sequence, and so

${\displaystyle TP|_{M}=TM\oplus T^{\perp }M.}$

Relative to this splitting, the Levi-Civita connection ∇′ of P decomposes into tangential and normal components. For each X ∈ TM and vector field Y on M,

${\displaystyle \nabla '_{X}Y=\top (\nabla '_{X}Y)+\bot (\nabla '_{X}Y).}$

Let

${\displaystyle \nabla _{X}Y=\top (\nabla '_{X}Y),\quad \alpha (X,Y)=\bot (\nabla '_{X}Y).}$

Gauss' formula[5] now asserts that ∇X 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.

An immediate corollary is the Gauss equation. For X, Y, Z, W ∈ TM,

${\displaystyle \langle R'(X,Y)Z,W\rangle =\langle R(X,Y)Z,W\rangle +\langle \alpha (X,Z),\alpha (Y,W)\rangle -\langle \alpha (Y,Z),\alpha (X,W)\rangle }$

where R′ is the Riemann curvature tensor of P and R is that of M.

The Weingarten equation is an analog of the Gauss formula for a connection in the normal bundle. Let X ∈ TM and ξ a normal vector field. Then decompose the ambient covariant derivative of ξ along X into tangential and normal components:

${\displaystyle \nabla '_{X}\xi =\top (\nabla '_{X}\xi )+\bot (\nabla '_{X}\xi )=-A_{\xi }(X)+D_{X}(\xi ).}$

Then

1. Weingarten's equation: ${\displaystyle \langle A_{\xi }X,Y\rangle =\langle \alpha (X,Y),\xi \rangle }$
2. DX 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 TM and TM. In particular, they defined the covariant derivative of α:

${\displaystyle ({\tilde {\nabla }}_{X}\alpha )(Y,Z)=D_{X}\left(\alpha (Y,Z)\right)-\alpha (\nabla _{X}Y,Z)-\alpha (Y,\nabla _{X}Z).}$

The Codazzi–Mainardi equation is

${\displaystyle \bot \left(R'(X,Y)Z\right)=({\tilde {\nabla }}_{X}\alpha )(Y,Z)-({\tilde {\nabla }}_{Y}\alpha )(X,Z).}$

Since every 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):

${\displaystyle L_{v}-M_{u}=L\Gamma ^{1}{}_{12}+M(\Gamma ^{2}{}_{12}-\Gamma ^{1}{}_{11})-N\Gamma ^{2}{}_{11}}$
${\displaystyle M_{v}-N_{u}=L\Gamma ^{1}{}_{22}+M(\Gamma ^{2}{}_{22}-\Gamma ^{1}{}_{12})-N\Gamma ^{2}{}_{12}}$

Derivation of classical equations

Consider a parametric surface in Euclidean 3-space,

${\displaystyle \mathbf {r} (u,v)=(x(u,v),y(u,v),z(u,v))}$

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 ru and rv are linearly independent. Complete this to a basis{ru,rv,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.

${\displaystyle {\mathbf {r}}_{uu}=\Gamma ^{1}{}_{11}{\mathbf {r}}_{u}+\Gamma ^{2}{}_{11}{\mathbf {r}}_{v}+L{\mathbf {n}}}$
${\displaystyle {\mathbf {r}}_{uv}=\Gamma ^{1}{}_{12}{\mathbf {r}}_{u}+\Gamma ^{2}{}_{12}{\mathbf {r}}_{v}+M{\mathbf {n}}}$
${\displaystyle {\mathbf {r}}_{vv}=\Gamma ^{1}{}_{22}{\mathbf {r}}_{u}+\Gamma ^{2}{}_{22}{\mathbf {r}}_{v}+N{\mathbf {n}}}$

Clairaut's theorem states that partial derivatives commute:

${\displaystyle \left({\mathbf {r}}_{uu}\right)_{v}=\left({\mathbf {r}}_{uv}\right)_{u}}$

If we differentiate ruu with respect to v and ruv with respect to u, we get:

${\displaystyle \left(\Gamma ^{1}{}_{11}\right)_{v}{\mathbf {r}}_{u}+\Gamma ^{1}{}_{11}{\mathbf {r}}_{uv}+\left(\Gamma ^{2}{}_{11}\right)_{v}{\mathbf {r}}_{v}+\Gamma ^{2}{}_{11}{\mathbf {r}}_{vv}+L_{v}{\mathbf {n}}+L{\mathbf {n}}_{v}}$ ${\displaystyle =\left(\Gamma ^{1}{}_{12}\right)_{u}{\mathbf {r}}_{u}+\Gamma ^{1}{}_{12}{\mathbf {r}}_{uu}+\left(\Gamma _{12}^{2}\right)_{u}{\mathbf {r}}_{v}+\Gamma ^{2}{}_{12}{\mathbf {r}}_{uv}+M_{u}{\mathbf {n}}+M{\mathbf {n}}_{u}}$

Now substitute the above expressions for the second derivatives and equate the coefficients of n:

${\displaystyle M\Gamma ^{1}{}_{11}+N\Gamma ^{2}{}_{11}+L_{v}=L\Gamma ^{1}{}_{12}+M\Gamma ^{2}{}_{12}+M_{u}}$

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 + k)-dimensional smooth manifold P. Let ${\displaystyle e_{1},e_{2},\ldots ,e_{k}}$ be a local orthonormal frame of vector fields normal to M. Then we can write,

${\displaystyle \alpha (X,Y)=\sum _{j=1}^{k}\alpha _{j}(X,Y)e_{j}}$

If, now, ${\displaystyle E_{1},E_{2},\ldots ,E_{m}}$ is a local orthonormal frame (of tangent vector fields) on the same open subset of M, then we can define the mean curvatures of the immersion by

${\displaystyle H_{j}=\sum _{i=1}^{m}\alpha _{j}(E_{i},E_{i})}$

In particular, if M is a hypersurface of P, i.e. ${\displaystyle k=1}$, then there is only one mean curvature to speak of. The immersion is called minimal if all the ${\displaystyle H_{j}}$ 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 ${\displaystyle 1/m}$.

We can now write the Gauss–Codazzi equations as

${\displaystyle \langle R'(X,Y)Z,W\rangle =\langle R(X,Y)Z,W\rangle +\sum _{j=1}^{k}\alpha _{j}(X,Z)\alpha _{j}(Y,W)-\alpha _{j}(Y,Z)\alpha _{j}(X,W)}$

Contracting the ${\displaystyle Y,Z}$ components gives us

${\displaystyle Ric'(X,W)=Ric(X,W)+\sum _{j=1}^{k}\langle R'(X,e_{j})e_{j},W\rangle +\left(\sum _{i=1}^{m}\alpha _{j}(X,E_{i})\alpha _{j}(E_{i},W)\right)-H_{j}\alpha _{j}(X,W)}$

Observe that the tensor in parentheses is symmetric and nonnegative-definite in ${\displaystyle X,W}$. Assuming that M is a hypersurface, this simplifies to

${\displaystyle Ric'(X,W)=Ric(X,W)+\langle R'(X,n)n,W\rangle +\left(\sum _{i=1}^{m}h(X,E_{i})h(E_{i},W)\right)-Hh(X,W)}$

where ${\displaystyle n=e_{1}}$ and ${\displaystyle h=\alpha _{1}}$ and ${\displaystyle H=H_{1}}$. In that case, one more contraction yields,

${\displaystyle R'=R+2Ric'(n,n)+\|h\|^{2}-H^{2}}$

where ${\displaystyle R'}$ and ${\displaystyle R}$ are the respective scalar curvatures, and

${\displaystyle \|h\|^{2}=\sum _{i,j=1}^{m}h(E_{i},E_{j})^{2}}$

If ${\displaystyle k>1}$, the scalar curvature equation might be more complicated.

We can already use these equations to draw some conclusions. For example, any minimal immersion[6] into the round sphere ${\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{m+k+1}^{2}=1}$ must be of the form

${\displaystyle \triangle x_{j}+\lambda x_{j}=0}$

where ${\displaystyle j}$ runs from 1 to ${\displaystyle m+k+1}$ and

${\displaystyle \triangle =\sum _{i=1}^{m}\nabla _{E_{i}}\nabla _{E_{i}}}$

is the Laplacian on M, and ${\displaystyle \lambda >0}$ is a positive constant.

Notes

1. ^
2. ^ (Kline 1972, p. 885).
3. ^
4. ^
5. ^ Terminology from Spivak, Volume III.
6. ^ Takahashi 1966

References

• Bonnet, Ossian (1867), "Memoire sur la theorie des surfaces applicables sur une surface donnee", Journal de l'École Polytechnique, 25: 31–151
• do Carmo, Manfredo Perdigao (1994), Riemannian Geometry, Francis Flaherty
• Codazzi, Delfino (1868–1869), "Sulle coordinate curvilinee d'una superficie dello spazio", Ann. Math. Pura applicata, 2: 101–19
• Gauss, Carl Friedrich (1828), "Disquisitiones Generales circa Superficies Curvas" [General Discussions about Curved Surfaces], Comm. Soc. Gott. (in Latin), 6 ("General Discussions about Curved Surfaces")
• Ivanov, A.B. (2001), "Peterson–Codazzi equations", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Kline, Morris (1972), Mathematical Thought from Ancient to Modern Times, Oxford University Press, ISBN 0-19-506137-3
• Mainardi, Gaspare (1856), "Su la teoria generale delle superficie", Giornale dell' Istituto Lombardo, 9: 385–404
• Peterson, Karl Mikhailovich (1853), Über die Biegung der Flächen, Doctoral thesis, Dorpat University.
• Takahashi, Tsunero (1966), "Minimal Immersions of Riemannian Manifolds", Journal of the Mathematical Society of Japan