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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 Hodgede 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 Hodgede 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 Hodgede 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.
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