Let $(M,g)$ a Riemannian manifold. I'm having difficulties for find the expression for $Ric(\nabla f ,\nabla h)$, where $f,g \in C^{3}(M)$.
My notes says that $Ric(\nabla f ,\nabla h)=g^{ij}g^{kl}\nabla_{k}\nabla_{i}f\nabla_{l}\nabla_{j} h$. in a smooth coordinate system $(x^{i})$.
In my manipulation
- $Ric(\nabla f ,\nabla h)=Ric(g^{ij}\nabla_{i}f\partial_{j},g^{kl}\nabla_{l}h\partial_{k})=g^{ij}g^{kl}\nabla_{i}f\nabla_{l}hRic(\partial_{j},\partial_{k})=g^{ij}g^{kl}\nabla_{i}f\nabla_{l}hR_{jk}$ (*)
The affirmation is wrong? I don't know as continue (*)