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(adsbygoogle = window.adsbygoogle || []).push({});

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

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!

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# Generalized Gauss-Codazzi equations

Can you offer guidance or do you also need help?

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