I am reading the article Some Geometric Calculations on Wasserstein Spaces of John Lott and there is this covariant index in the covariant derivative: $\nabla^i$. And I don't quite understand it.

In some place he establishes the equality ($\phi_i$ and $\rho$ being smooth functions on $M$):

$$\int_M\langle\nabla\phi_1,\nabla\phi_2\rangle\rho d\operatorname{vol}_M=-\int_M\phi_1\nabla^i(\rho\nabla_i\phi_2)d\operatorname{vol}_M.$$

In another part of the text he writes $\nabla^i f$ for $f$ smooth. I think he is using the musical isomorphisms some way to define it, but I'm not sure how and I can't see how the above equation takes place.

It is worth mentioning that he applies $\nabla^i$ on functions, not on 1-forms. I'd like to understand the identity as well as to understand explicitly how $\nabla^i$ operates, since I want to caculate it on specific functions, such as $f_n(x)=e^{Inx}$, with $I$ being the imaginary unit.

An example of explicit calculation of $\nabla^i$ for an explicit function would be nice.

iscalled musical isomorphism. The differential (i.e. exterior derivative) of a function $f$ is a 1-form $df$, and the musical isomorphism defined by the metric $g$ turns this into the vector field $\nabla f.$ You can read for instance in John Lee's "Introduction to Riemannian manifolds" or most other introductory books on Riemannian geometry $\endgroup$3more comments