Von Neumann index theorems for manifolds with boundary
type: Item
instance of: scholarly article
title: Von Neumann index theorems for manifolds with boundary
zbMATH Open document ID: 0787.58040
author: Mohan Ramachandran
published in: Journal of Differential Geometry
publication date: 1994-05-16T00:00:00Z
review text: Some extensions of the notions involved in the well-known Atiyah-Singer index theorem have been given by Atiyah (the definition of the real- valued index of the elliptic differential operator on infinite Galois coverings of closed manifolds, equivariant with respect to the Galois group, by using the von Neumann generalized dimension), by Cheeger-Gromov (the generalized eta invariants for the infinite Galois coverings), by the author and G. Peric (the eta invariants for foliations). In the present paper the author proves the following results concerning the geometric Dirac operators and the index theorem for foliations. Theorem 1.1. Let \(D\) be a Dirac operator on a compact manifold \(M\) with boundary, acting on the sections of a graded Clifford module \(S\), with grading operator \(\varepsilon\). Assume that the data \((D,S,\varepsilon)\) have a product structure near the boundary. Let \(\widetilde{M}\) be a Galois covering with Galois group \(\Gamma\). Let \((\widetilde{D},\widetilde{S},\widetilde{\varepsilon})\) be the lift of the data \((D,S,\varepsilon)\) to \(\widetilde{M}\). Let \(\widetilde{B}\) be the Atiyah-Patodi-Singer boundary condition associated to \(\widetilde{D}\). Then the \(\Gamma\) index of the Breuer-Fredholm operator \(\widetilde{D}\) with boundary condition \(\widetilde{B}\) is given by \[ \text{ind}_ \Gamma(\widetilde{D}) = \int_ M\text{ch}(\sigma_ D)Td(M) - {\eta_ \Gamma(0) + h\over 2}, \] where \(\eta_ \Gamma(0)\) is the \(\Gamma\)-eta invariant on the boundary of \(\widetilde{M}\). Theorem 1.2. Let \((M,{\mathcal F})\) be a compact foliated manifold with boundary, with the foliation \(\mathcal F\) transverse to the boundary. Let \(D_{\mathcal F}\) be a leafwise Dirac operator acting leafwise on a Clifford module \(S\) over the tangent bundle \(T{\mathcal F}\) to the foliation. Assume that \(S\) is graded with grading operator \(\varepsilon\). Let \(\nu\) be a holonomy invariant transverse measure for the foliation \(\mathcal F\). Further …
Mathematics Subject Classification ID: 58J20
zbMATH DE Number: 488226
zbMATH Keywords: Dirac operators
zbMATH Keywords: index theorem
zbMATH Keywords: foliations
Wikidata QID: Q115180143
reviewed by: Vasile Oproiu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214454297
OpenAlex ID: W2955546572
DOI: 10.4310/JDG/1214454297
description: scientific article; zbMATH DE number 488226