  In this survey we collect some recent results regarding the Lipschitz-Killing
curvatures (LKCs) of the excursion sets of random eigenfunctions on the
two-dimensional standard flat torus (arithmetic random waves) and on the
two-dimensional unit sphere (random spherical harmonics). In particular, the
aim of the present survey is to highlight the key role of integration by parts
formulae in order to have an extremely neat expression for the random LKCs.
Indeed, the main tool to study local geometric functionals of random waves on
manifold is to exploit their Wiener chaos decomposition and show that (often),
in the so-called high-energy limit, a single chaotic component dominates their
behavior. Moreover, reduction principles show that the dominant Wiener chaotic
component of LKCs of random waves' excursion sets at threshold level $u\ne0$ is
proportional to the integral of $H_2(f)$, f being the random field of interest
and $H_2$ the second Hermite polynomial. This will be shown via integration by
parts formulae.
