Spectrum and the fixed point sets of isometries. III
type: Item
instance of: scholarly article
zbMATH Open document ID: 0986.58011
author: Harold Donnelly
publication date: 2002-06-06T00:00:00Z
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 1670978
zbMATH Keywords: fixed point sets of isometries
zbMATH Keywords: Dirac operators
zbMATH Keywords: index theorem
zbMATH Keywords: spectrum
zbMATH Keywords: \(G\)-signature theorem
zbMATH Keywords: Hermitian vector bundle
MaRDI profile type: Publication
Recommended article: On isometries with finite spectrum
Recommended article: Q5289182
Recommended article: On \(G\)-sets and isospectrality
Recommended article: Q5447328
Recommended article: A fixed point approach to the stability of isometries
Recommended article: Partial isometries and a general spectral theorem
Recommended article: Fixed points of isometries on weakly compact convex sets
Recommended article: Q3790361
Recommended article: On the isospectra and the isometries of the Aloff-Wallach spaces
title: Spectrum and the fixed point sets of isometries. III
review text: Remark added in 2000. This paper was written in 1976 and has remained unpublislied for twenty five years. For a survey of many subsequent developments, the reader should consult the textbook by \textit{N. Berline}, \textit{E. Getzler}, and \textit{M. Vergne}, entitled ``Heat kernels and Dirac Operators'' (1992; Zbl 0744.58001).NEWLINENEWLINENEWLINEFrom the introduction: Let \(M\) be a compact Riemannian manifold of dimension \(d\) and \(f:M\to M\) an isometry with fixed point set \(\Omega\). \(\Omega\) is the disjoint union of closed connected submanifolds \(N\) of dimension \(n\). If \(\Delta\) denotes the Laplace operator of \(M\) with eigenvalues \(\lambda\), then \(f\) induces linear maps \(f^\#_\lambda\) on the eigenspaces of \(\Delta\). For each \(N\in\Omega\) and \(a\in N\), \(f\) induces an \(0(d-n)\) action \(A:(T_aN)^\perp \to(T_aN)^\perp\) on the fiber of the normal bundle \((TN)^\perp\) to \(N\) at \(a\). \((I-A)\) is invertible and we denote \(B=(I-A)^{-1}\). NEWLINENEWLINENEWLINEIn part I of this paper [\textit{H. Donnelly}, Math. Ann. 224, 161-170 (1976; Zbl 0327.53031)] the following asymptotic expansion was derived: NEWLINE\[NEWLINE\sum_\lambda \text{Tr} (f_\lambda^\#) e^{-t\lambda} \sim\sum_{N\in \Omega}(4 \pi t)^{-{n\over 2}}\sum^\infty_{k=0} t^k\int_N b_k(f,a)d vol_N(a).NEWLINE\]NEWLINE It was also shown that the \(b_k(f,a)\) have a simple description using invariant theory. This suggested a new proof of the Atiyah-Singer-Lefschetz formulas for compact group actions. In Part II [\textit{H. Donnelly} and \textit{V. K. Patodi}, Topology 16, 1-11 (1977; Zbl 0341.53023)] we derived such a proof of the \(G\)-signature theorem of Atiyah-Singer.NEWLINENEWLINENEWLINEIn this paper (Part III) we present the additional details needed to deduce the \(G\)-signature theorem with coefficients in a Hermitian vector bundle. While the methods of Part II clearly generalize, it is nevertheless of some interest to write down these results in a careful manner. In part…
description: scientific article; zbMATH DE number 1670978