# Generalized Gauss-Codazzi equations

1. Oct 18, 2013

### 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!