Ricci flow on surfaces : step in proof

Question

I am trying to realize the paper of richard hamilton's ricci flow on surfaces from the book of benett chow's Ricci flow : An Introduction.Here Hamilton denoted the trace free part of the Hessian of the potential $f$ of the curvature by $$\ M = \nabla \nabla f - \frac 12 \Delta f . g $$ Next taking divergence of $M$ we have $$(div M)_i = \nabla ^j M_{ji}=\nabla _j\nabla_i\nabla^jf-\frac 12 \nabla_i\nabla_j\nabla^jf=R_{ik}\nabla^kf+\frac 12\nabla_i\Delta f=\frac 12(R\nabla_if+\nabla_iR)$$....But in this calculation I can not find what the term $\nabla^j$ means.$\nabla_j$ is covariant derivative...But what about $\nabla^j$...Please anyone help me to understand the calculations...one more here $R_{ik}=\frac R2 g_{ik}$...

Answer 1

This is a convention in differential geometry: $$\nabla^{j} = g^{ij}\nabla_i=\sum_{i=1}^{n}g^{ij}\nabla_i$$ where n is the dimension of the manifold.