A signature theorem for disk bundles and the eta invariant
type: Item
instance of: scholarly article
title: A signature theorem for disk bundles and the eta invariant
zbMATH Open document ID: 0786.58037
DOI: 10.1016/0926-2245(93)90026-W
author: Sanjay Kumar Tiwari
published in: Differential Geometry and its Applications
publication date: 1994-04-25T00:00:00Z
review text: The author applies the signature theorem for compact Riemannian manifolds with boundary [\textit{M. F. Atiyah, V. K. Patodi} and \textit{I. M. Singer}, Math. Proc. Camb. Philos. Soc. 77, 43-69 (1975; Zbl 0297.58008) and \textit{P. B. Gilkey}, Adv. Math. 15, 334-360 (1975; Zbl 0306.53042)] to disc bundles over compact manifolds. Let \(E\) be an oriented real vector bundle of rank 2 over a compact oriented Riemannian manifold \(M\) of dimension \(4k-2\). It is assumed that the Riemannian metric on the total space of \(E\) is induced by a fibre metric \(h\) on \(E\) and a connection on \(E\) compatible with \(h\). The signature of the unit disc bundle of \(E\) is computed as a sum of an integral over \(M\) of a characteristic polynomial \(L_ k(E,M)\) and \(\lim_{r\to 0}\eta(S_ r(E))\). Here \(L_ k(E,M)\) is a homogeneous polynomial of degree \(2k-1\), which can be expressed explicitly in terms of the coefficients of the Hirzebruch \(L\)-polynomial, the Euler class of \(E\) and the Pontryagin classes of \(M\), and \(\eta(S_ r(E))\) is the eta-invariant of the circle bundle of radius \(r\). This result is used to compute the signature of \(D(E)\) of holomorphic vector bundles over compact complex manifolds. When \(M\) is an odd dimensional Hodge manifold and \(E\) is a positive holomorphic line bundle the signature of \(D(E)\) is expressed by the bigraded Betti numbers of the base.
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 32L05
zbMATH DE Number: 440261
zbMATH Keywords: index theory
zbMATH Keywords: manifolds with boundary
zbMATH Keywords: disc bundles
zbMATH Keywords: compact manifolds
zbMATH Keywords: eta-invariant
zbMATH Keywords: holomorphic vector bundles
reviewed by: K. H. Mayer
MaRDI profile type: Publication
Wikidata QID: Q115362686
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description: scientific article; zbMATH DE number 440261