On orthogonal direct sum decomposition of de Rham complexes for warped products of Riemannian manifolds
type: Item
instance of: scholarly article
title: On orthogonal direct sum decomposition of de Rham complexes for warped products of Riemannian manifolds
zbMATH Open document ID: 0944.58001
DOI: 10.1007/BF02677519
author: V. I. Kuz'minov
author: I. A. Shvedov
published in: Siberian Mathematical Journal
publication date: 2000-04-13T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 58A10
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 1431657
zbMATH Keywords: exterior derivative
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: warped product
zbMATH Keywords: de Rham complex
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02677519
OpenAlex ID: W2067483383
description: scientific article; zbMATH DE number 1431657