Let us consider the Ricci identity defining the Riemann tensor as measuring the lack of commutation of two successive covariant derivatives with respect to the connection associated with ‘s metric
By using as the normal vector , we find that
We now project this tensor twice onto , here the indices and , and once along , here the index . Using the previous topic, we find that
We now denote by the following vector . It is clear that is a tangent vector.
This is clear since
By extending the second fundamental on given by onto , we find that for all vectors .
Again, this is clear since
Thus, in components, we have
We now calculate in terms of the lapse function . Thanks to , we find that
Therefore, we can write
Using this, we can calculate as follows.
For any normal vector , say , we obtain the following Lie derivative
Since , we find that
We now project the above equation onto and thanks to the fact that tangents to because does, we arrive at
Thus, we have proved that
This is the so-called Ricci equation. Note that we can write
since has no torsion.