Gauss–Codazzi equations


In Riemannian geometry and pseudo-Riemannian geometry, the Gauss-Codazzi equations are fundamental formulas which link together the induced metric and second fundamental form of a submanifold of a 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, 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. 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 and Delfino Codazzi, who independently derived the result, although it was discovered earlier by Karl Mikhailovich Peterson.

Formal statement

Let be an n-dimensional embedded submanifold of a Riemannian manifold P of dimension. 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:
The metric splits this short exact sequence, and so
Relative to this splitting, the Levi-Civita connection of P decomposes into tangential and normal components. For each and vector field Y on M,
Let
The Gauss formula now asserts that is the Levi-Civita connection for M, and 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,
where 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 and a normal vector field. Then decompose the ambient covariant derivative of along X into tangential and normal components:
Then
  1. Weingarten's equation:
  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 T⊥M. In particular, they defined the covariant derivative of :
The Codazzi–Mainardi equation is
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 :
The Gauss formula, depending on how one chooses to define the Gaussian curvature, may be a tautology

Derivation of classical equations

Consider a parametric surface in Euclidean 3-space,
where the three component functions depend smoothly on ordered pairs 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, 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.
Clairaut's theorem states that partial derivatives commute:
If we differentiate ruu with respect to v and ruv with respect to u, we get:
Now substitute the above expressions for the second derivatives and equate the coefficients of n:
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 -dimensional smooth manifold P. Let be a local orthonormal frame of vector fields normal to M. Then we can write,
If, now, is a local orthonormal frame on the same open subset of M, then we can define the mean curvatures of the immersion by
In particular, if M is a hypersurface of P, i.e., then there is only one mean curvature to speak of. The immersion is called minimal if all the 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.
We can now write the Gauss–Codazzi equations as
Contracting the components gives us
Observe that the tensor in parentheses is symmetric and nonnegative-definite in. Assuming that M is a hypersurface, this simplifies to
where and and. In that case, one more contraction yields,
where and are the respective scalar curvatures, and
If, the scalar curvature equation might be more complicated.
We can already use these equations to draw some conclusions. For example, any minimal immersion into the round sphere must be of the form
where runs from 1 to and
is the Laplacian on M, and is a positive constant.