Some Problems Regarding the Spectra of Hodge-de Rham Operators : The smooth and continuous dependence on the Riemannian metric of the eigenvalues of the Hodge-de Rham operators and its consequences

Spectral geometry deals with the survey of these natural, differential operators' spectrums and among other things it tries to emphasize geometrical and topological properties of a manifold that can be recuperated from the spectrums. The present work is going to approach several issues referring to the spectrums of Hodge-de Rham operators on closed Riemannian manifolds. The author of this paper is going to discuss the continuous dependence on the Riemannian metrics on a smooth and closed differential manifold of the eigenvalues of the Hodge-de Rham operators and its restrictions regarding the exact, differential form spaces and consequences of such feature. Moreover, by using J. Wenzelburger's idea [80], [81], we are going to prove that the eigenvalues of the Hodge-de Rham operators even smoothly depend on the Riemannian metrics on a smooth, closed, differential manifold if the Frchet smooth manifold canonical structure is taken into consideration in the space of all Riemannian metrics with such a manifold.