What is the meaning of the symbol $\nabla^k$?

Question

In T.Aubin's book, a course in differential geometry, he write the formula $\Delta f=-\nabla^k\nabla_kf$ on a Riemannian manifold, but he never define the symbol $\nabla^k$. It seems that the notation is not the kth covariant derivative. So my question is: Does any one know what is the meaning of the symbol? Thank you very much and any reference is welcome.

Answer 1

It looks like here on functions $\nabla^k \nabla_k f$ means $g^{jk}\nabla_j\nabla_k f$ where $\nabla_j\nabla_k f = \nabla_{\partial_j}\nabla_{\partial_k}f = (\nabla \nabla f)_{jk}$. So for an $r$-tensor $\alpha_{i_1 \cdots i_r}$, we could lift an index on the $(r+1)$-tensor $\nabla_k \alpha_{i_1 \cdots i_r} = (\nabla \alpha)_{ki_1 \cdots i_r}$ and write $\nabla^k \alpha_{i_1 \cdots i_r}$ for $g^{jk}\nabla_j \alpha_{i_1 \cdots i_r}$.