The index of elliptic operators on manifolds with conical points
type: Item
instance of: scholarly article
title: The index of elliptic operators on manifolds with conical points
zbMATH Open document ID: 0951.58026
published in: Selecta Mathematica. New Series
publication date: 2000-06-07T00:00:00Z
full work available at URL: urn:nbn:de:kobv:517-opus-25096
review text: The authors offer a new perspective on the index theory of elliptic pseudodifferential operators on manifolds with isolated conical singularities. Their approach is based on the algebra of formal symbols, an example of deformation quantization whose role in index theory is due to \textit{B. V. Fedosov} [Trans. Moscow Math. Soc. 30, 159-240 (1976; Zbl 0349.58006); translation from Trudy Moskov. Mat. Obsc. 30, 159-241 (1974)]. In a departure from previous work on cone Mellin pseudodifferential operators, the authors choose variables in which a deleted neighborhood of the conical point is represented by an infinite cylindrical end. The index formula proven in this paper is given an attractive interpretation in the two-dimensional case. There the formula depends entirely on the principal interior symbol and on the conormal symbol. The formula decomposes as the sum of an Atiyah-Singer integral, an eta-invariant term, and a third term. Calculations with the Cauchy-Riemann operator show that all three terms can be nonzero. The paper's final section interprets the index formula in the context of a symmetry condition on the conormal symbol in arbitrary dimension.
reviewed by: Peter Haskell
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 58J40
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 58J28
Mathematics Subject Classification ID: 53D55
zbMATH DE Number: 1422753
zbMATH Keywords: index theorem
zbMATH Keywords: elliptic pseudodifferential operator
zbMATH Keywords: manifold with singularities
author: Boris V. Fedosov
author: B.-W. Schulze
author: N. N. Tarkhanov
MaRDI profile type: Publication
OpenAlex ID: W2074408992
DOI: 10.1007/S000290050054
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description: scientific article; zbMATH DE number 1422753