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If $M$ is a s-R(semi-Riemannian) submanifold of a s-R manifold $\overline{M}$ the function $R^{\perp}:\mathfrak{X}(M)\times\mathfrak{X}(M)\times\mathfrak{X}(M)^\perp\rightarrow\mathfrak{X}(M)^\perp$ given by

$$R_{VW}^\perp X=D_{[V,W]}^\perp X-[D_V^\perp,D_W^\perp]X $$

is called the normal curvature tensor of $M$. The question is, how can I prove that it satisfies the Ricci equation:

$$\langle R_{VW}^\perp X,Y\rangle=\langle \bar{R}_{VW} X,Y\rangle+\langle\widetilde{II}(V,X),\widetilde{II}(W,Y)\rangle-\langle\widetilde{II}(V,Y),\widetilde{II}(W,X)\rangle,$$

where $X,Y\in\mathfrak{X}^\perp(M)$?

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    This is [Gauss-Codazzi equation][1] in the Riemannian case. Have you ever try to look at the proof of the Riemannian case to see if everything goes through for the semi-Riemannian case? [1]: http://en.wikipedia.org/wiki/Gauss%E2%80%93Codazzi_equations2011-12-13

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