covariant derivative of ricci scalar

Question

How to derivate (covariant derivative) the expressions which is function of Ricci scalar? Also, if R is Ricci scalar, what would be ∇i∇j F(R) ?, where ∇i is covariant derivative.

Answer 1

I'm assuming $F$ is a scalar-valued function with no dependence on location within the manifold (though $R$ of course may vary). I believe the expression you are looking for would then be: $$\nabla_i\nabla_j F(R) = \nabla_i (\partial_j F(R)) = \nabla_i(F'(R)\partial_jR) = \partial_i(F'(R)\partial_jR)-\Gamma^k_{ij}F'(R)\partial_kR$$ $$\nabla_i\nabla_j F(R) = F''(R)\partial_iR\partial_jR+F'(R)(\partial_i\partial_jR - \Gamma^k_{ij}\partial_kR)$$