An index theorem for open manifolds whose ends are warped products
type: Item
instance of: scholarly article
title: An index theorem for open manifolds whose ends are warped products
zbMATH Open document ID: 1019.58007
DOI: 10.1007/BF02941677
author: P. D. Baier
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2003-02-13T00:00:00Z
review text: The author gives a careful analysis of the index theory of Dirac operators on a class of even-dimensional, complete, noncompact, spin manifolds. A manifold is in the class if it has a compact subset whose complement is topologically a cylinder and metrically a warped product (not necessarily cylindrical). The author provides some conditions sufficient to guarantee that the Dirac operator on such a manifold is Fredholm and some conditions that imply that the operator is not Fredholm. Whether or not the Dirac operator is Fredholm, it has a finite \(L^{2}\)-index, defined to be the dimension of the kernel of its restriction to positive \(L^{2}\) spinors minus the dimension of the kernel of its restriction to negative \(L^{2}\) spinors. In many cases the author provides a formula for the \(L^{2}\)-index. This formula is based on the extension of the Atiyah-Patodi-Singer index theorem due to \textit{G. Grubb} [Commun. Partial Differ. Equations 17, 2031-2077 (1992; Zbl 0773.58025)] and \textit{P. B. Gilkey} [Adv. Math. 102, 129-183 (1993; Zbl 0798.58071)]. Under some conditions the \(L^{2}\)-index is guaranteed to be zero, but in general the index formula includes two nonlocal terms.
reviewed by: Peter Haskell
Mathematics Subject Classification ID: 58J20
zbMATH DE Number: 1868560
zbMATH Keywords: index theory
zbMATH Keywords: Dirac operator
zbMATH Keywords: warped product
zbMATH Keywords: \(L^2\)-index
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1868560