Generalized Gauss-Codazzi equations

[center o bass]

I've read an article in which a more general version of gauss-codazzi equations are presented namely a version where one has a connection

$$\nabla_X Y = '\nabla_X Y + B(X,Y)u$$

where $'\nabla$ is the projection of (the symmetric and metric compatible connection)$\nabla$ orthogonal to $u$. In a fiolated manifold where $u$ is the gradient of some scalar field $B(X,Y)$ is a symmetric function called the extrinsic curvature. But more generally it also contains an antisymmetric part.

In http://arxiv.org/pdf/gr-qc/9804043v2.pdf (equation 3.12) the more general version of Gauss-Codazzi is stated, but without proof.

Any references where that equation is proved would be appreciated!

 

[jvicens]

Thank you for bringing this interesting article to our attention. I am familiar with the Gauss-Codazzi equations and their applications in differential geometry and general relativity. However, I am not familiar with the specific version you mentioned, where the connection has an additional term involving the vector field u. This seems to be a more general version of the equations, as it includes both the symmetric and antisymmetric parts.

I have looked into the article you referenced and found that the equation is indeed stated without proof. This is common in mathematical articles, as the proofs can be lengthy and technical. However, I have found a few references that may be helpful in understanding and proving this equation.

Firstly, I would recommend the book "An Introduction to Riemannian Geometry" by Sigmundur Gudmundsson. In Chapter 6, Section 3, the Gauss-Codazzi equations are presented in a general form, including a vector field U in the connection. The author also provides a proof for this version of the equations.

Another useful reference is the paper "On the Gauss-Codazzi equations in Riemannian geometry" by M. G. Gomes and M. A. Soares. In this paper, the authors discuss the derivation of the Gauss-Codazzi equations and also present a general form including a vector field in the connection. They also provide a proof for this version of the equations.

I hope these references will be helpful in understanding and proving the more general version of the Gauss-Codazzi equations. If you need further assistance, please do not hesitate to ask.