Ricci and Scalar Curvatures
The Ricci curvature tensor is essentially the unique nontrivial way of contracting the Riemann tensor:
The Ricci tensor is symmetric.
By the contracting relations on the Christoffel symbols, we have
The scalar curvature is the trace of the Ricci curvature,
The "gradient" of the scalar curvature follows from the Bianchi identity (proof):