I was reading from the book of T. Aubin Some Nonlinear Problems in Riemannian Geometry some basic aspects about the laplacian defined on a manifold. Concretely, regarding its eigenvalues I have seen the next proof of the fact that all of them are nonnegative.
I know this proof in $\mathbb{R}^n$, but here it is not quite clear for me the meaning $\nabla^v f$ and $\nabla_vf$ and why the product has to be positive. Are they related with the metric that we have on $g$? To better understand, if we take for example $M=\mathbb{R}^2/\mathbb{Z}^2$ (flat torus), how are they defined?