Let $\Sigma \subset M$ be a hypersurface.
Gauss-Codazzi equation:
$$^\Sigma \text{Rc}_{ij} =\ ^M \text{Rc}_{ij} -\ ^M \text{Rm}(\partial_i,\nu,\nu,\partial_j) + H A_{ij} - A_{il} A^l_j \tag{1}$$ or
$$^M \text{Rm}(\partial_i,\nu,\nu,\partial_j) =\ ^M \text{Rc}_{ij} -\ ^\Sigma \text{Rc}_{ij} + H A_{ij} - A_{il} A^l_j \tag{2}$$
where
$^\Sigma \text{Rc}_{ij}$ is the Ricci tensor on $\Sigma$
$^M \text{Rm}$ is the Riemann tensor on manifold $M$
$\nu$ is the unit normal vector to $\Sigma$
$H$ is the mean curvature
$A_{ij}$ is the second fundamental form
Question:
How to calculate $^M \text{Rm}(\partial_i,\nu,\nu,\partial_j)$?
Here I have
$$^M \text{Rm}(\partial_i,\nu,\nu,\partial_j) =\ ^M \text{Rc}^T_{ij} + G(\nu,\nu)g_{ij} \\ ^M \text{Rm}(\partial_i,\nu,\nu,\partial_j) =\ ^MR_{ij}^T + \bigg(\ ^{M}\text{Rc}(\nu,\nu) - \frac{^MR}{2}\ \cdot g(\nu,\nu) \bigg) g_{ij} \tag{3}$$
where $G =\ ^M \text{Rc} - \frac{1}{2}\ ^M \text{Sc} \cdot g$ is the Einstein tensor. Since $g(\nu,\nu) = |\nu|^2=1$, then (3) becomes
$$^M \text{Rm}(\partial_i,\nu,\nu,\partial_j) =\ ^MR_{ij}^T + \bigg(\ ^{M}\text{Rc}(\nu,\nu) - \frac{^MR}{2} \bigg) g_{ij} \tag{4}$$
But I am afraid I am just repeating the equation (2).
Please help. Thank you.