Reference: Here
Let $\Sigma \subset M$ be an embedded surface in a manifold $M$.
At Section 2 page 11, there is $T_{ij}$ which is defined by
$$T_{ij} =\ ^M \text{Rm}(\partial_i,\nu,\nu,\partial_j) =\ ^M \text{Rc}^T_{ij} + G_(\nu,\nu)\gamma_{ij} $$
and $$G=\ ^M \text{Rc} - \frac{1}{2}\ ^M \text{Sc} \cdot g$$ is the Einstein tensor.
Questions:
What is the definition and mathematical expression of $^M \text{Rc}^T_{ij}$?
From the Gauss equation we have
$$^\Sigma \text{Rc}_{ij} =\ ^M \text{Rc}_{ij} -\ ^M \text{Rm}(\partial_i,\nu,\nu,\partial_j) + H A_{ij} - A_{il} A^l_j $$
Is this $^M \text{Rc}_{ij}$ the same with $^M \text{Rc}^T_{ij}$ mentioned above in Question 1 or are they different?
Thank you.