Let $(M^n,g)$ be a compact Riemannian manifold (without boundary). The symmetries of the curvature $R$ of (the Levi-Civita connection associated to) $g$ allow one to realise $R$ as a self-adjoint (with respect to the induced metric on $\Lambda^2(T^{\ast}M)$) operator $$\mathfrak{R} : \Lambda^2(T^{\ast} M) \to \Lambda^2(T^{\ast} M).$$
Question. Is there any relationship between the eigenvalues of $\mathfrak{R}$ and those of some Laplacian operator (e.g., Laplace-Beltrami, rough, etc.)?
