\(L_p\)-cohomologies of warped products of Lipschitz Riemannian manifolds
type: Item
instance of: scholarly article
title: \(L_p\)-cohomologies of warped products of Lipschitz Riemannian manifolds
zbMATH Open document ID: 0904.58001
DOI: 10.1007/BF02673913
author: K. V. Storozhuk
author: I. A. Shvedov
published in: Siberian Mathematical Journal
publication date: 1998-11-22T00:00:00Z
review text: Let \(X\) and \(Y\) be Lipschitz Riemannian manifolds of dimensions \(m\) and \(n\) and \(p\in [1,\infty]\). The main result of the article is the demonstration of the Künneth formula in the case when either \(Y\) is compact or its de Rham \(L_p\)-complex is decomposable. More exactly, the result can be described as follows. Assume measurable functions \(\rho,f\: X\to \mathbb R_+\) and \(\sigma\: Y\to \mathbb R_+\) to be locally bounded with locally bounded inverses. If the function \(f\) is bounded and the Banach complex \(\mathbf{L}_p=( L_p^j(Y,\sigma;E),d^j_{W_p}\mid j\in \mathbb Z)\) \((\mathbf{W}_p=( W_p^i(Y,\sigma;E)\mid i\in \mathbb Z))\) is decomposable, then the Künneth formula \[ H_p^k(X \times_f Y,\rho \sigma;E) \cong\oplus_{i+j=k} H_p^i( X,\rho f^{n/p-j};H_p^j(Y,\sigma;E)) \] holds (\(E\) is a Banach space). Similarly, if the function \(f\) is bounded and \(Y\) is compact, then the formula \[ H_p^k(X \times_f Y,\rho;\mathbb R) \cong\oplus_{i+j=k} H_p^i( X,\rho f^{p/n-j};H^j(Y;\mathbb R)) \] holds.
Mathematics Subject Classification ID: 58A10
Mathematics Subject Classification ID: 58A12
zbMATH DE Number: 1225832
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: Künneth formula
zbMATH Keywords: de Rham \(L_p\)-complex
Wikidata QID: Q115391907
MaRDI profile type: Publication
cites work: L//2 cohomology of warped products and arithmetic groups
cites work: Q3772804
full work available at URL: https://doi.org/10.1007/bf02673913
OpenAlex ID: W2058962765
description: scientific article; zbMATH DE number 1225832