On orthogonal direct sum decomposition of de Rham complexes for warped products of Riemannian manifolds
Let \( M = X \times_f Y \) be the product of smooth oriented Riemannian manifolds endowed with the metric \( g_M = g_X + f^2 (x) \cdot g_Y \), where \( g_X \) and \( g_Y \) are the Riemannian metrics on \( X \) and \( Y \) and \( f \) is a smooth positive function on \( X \). The main aim of the article under review consists in description of some sufficient conditions for decomposability of de Rham and Hilbert complexes \( \bigoplus_k L^k_2 (M, \sigma) \) of forms on \( M \) into orthogonal sums of their special subcomplexes. The direct summands of these decompositions orthogonally reduce the corresponding Laplace operators \( {\Delta}^k_M \) defined on the spaces of \(k\)-forms and are connected with normal and compact solvability of the weighted complexes associated with \( \bigoplus_k L^k_2 (Y, \sigma) \). Here \( \sigma \) is a weight function on \( Y \). The decomposability results obtained are applied to constructing some reduced cohomologies of de Rham and Hilbert complexes of the warped product \( M \). For some previous results of the authors, see [Sib. Math. J. 37, No. 2, 276-287 (1996; Zbl 0893.58004), Sib. Math. J. 38, No. 3, 492-506 (1997; Zbl 0870.58003), and Sib. Math. J. 34, No. 1, 73-82 (1993; Zbl 0844.58003)].
scientific article; zbMATH DE number 1431657