If $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is smooth and compactly supported, one has
$$\int |\Delta f(\mathbf{x})|^2\,d\mathbf{x} = \int \| Hf(\mathbf{x}) \|\_F^2\,d\mathbf{x}\,,$$
where $\Delta$ denotes the Laplacian, $H$ denotes the Hessian (matrix of second derivatives), and $\| \cdot\|\_F^2$ denotes the Frobenius norm squared (the sum of squares of matrix entries). What can be said about compact Lie groups?


Specifically, I have two questions about this phenomenon:


*Nomenclature*: In this case where all norms are $L^2$, does this equality have a generally accepted name? What about the case when other norms are introduced?


*Lie groups*: I would like to understand in what sense this generalizes to compact Lie groups. What can be said about smooth $f: G \rightarrow \mathbb{R}$ (or $\mathbb{C}$)?



Relationship between Laplacian and Hessian on compact Lie groups
This has nothing to do with Lie groups, I believe. Let $M$ be a Riemannian manifold. The Bochner formula on $1$-forms states that
$$\nabla^\* \nabla \omega = (d \delta + \delta d)\omega - \mathrm{Ric}\,\omega.$$
Hence we have for any compactly supported function $f$ (writing round brackets for the $L^2$ scalar product) 
$$\|Hf\|\_{L^2}^2 = \|\nabla d f\|\_{L^2}^2 = (\nabla^\* \nabla d f, df) = ((d \delta + \delta d - \mathrm{Ric})d f, df ) = (d\delta d f, d f) - (\mathrm{Ric}\, df, df)$$
Using that on functions, $\Delta f = \delta d f$, this is equal to
$$(d \Delta f, df) - (\mathrm{Ric}\, df, df) = \|\Delta f\|\_{L^2}^2 - (\mathrm{Ric}\, df, df)$$
This is a standard formula of Riemannian geometry.


