Issue:

SSRSMI, Number 2, Volume XIX

Section:

Volume 19, Number 2

Abstract:

Let M be an oriented, closed and smooth manifold of dimension n, A^k(M) the space of smooth differential forms on M, and Îœ(M) the space of all Riemannian metrics on M, endowed with the canonical structure of smooth Fréchet manifold. Using an idea of J. Wenzelburger [10], [11], we prove that the eigenvalues of the Hodge-de Rham operator Δ^(k): A^k(M) → A^k(M) depend smoothly on the Riemannian metric g ∈ Îœ(M), for each k ∈ {1,2,…n}. Minimax principle (see Theorem 2.2 of M. Craioveanu, M. Puta and Th. M. Rassias [5], p. 286) imply the smoothly dependence on Riemannian metric of eigenvalues of Hodge-de Rham operators and of restrictions of these operators on spaces of differential exact forms, respectively of coexact forms. It is also shown that some Hodge-de Rham decompositions smoothly depend on the Riemannian metric.

Keywords:

Hodge-de Rham operator, eigenvalues, Hodge-de Rham decomposition.

Code [ID]:

SSRSMI200902V19S01A0001 [0003113]

Full paper:

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