NONLOCAL INVARIANTS IN INDEX THEORY

Authors
Citation
S. Rosenberg, NONLOCAL INVARIANTS IN INDEX THEORY, Bulletin, new series, of the American Mathematical Society, 34(4), 1997, pp. 423-433
Citations number
54
Categorie Soggetti
Mathematics, General",Mathematics
ISSN journal
02730979
Volume
34
Issue
4
Year of publication
1997
Pages
423 - 433
Database
ISI
SICI code
0273-0979(1997)34:4<423:NIIIT>2.0.ZU;2-2
Abstract
In its original form, the Atiyah-Singer Index Theorem equates two glob al quantities of a closed manifold, one analytic (the index of an elli ptic operator) and one topological (a characteristic number). Because it relates invariants from different branches of mathematics, the Inde x Theorem has many applications and extensions to differential geometr y, K-theory, mathematical physics, and other fields. This report focus es on advances in geometric aspects of index theory. For operators nat urally associated to a Riemannian metric on a closed manifold, the top ological side of the Index Theorem can often be expressed as the integ ral of local (i.e. pointwise) curvature expression. We win first discu ss these local refinements in 1, which arise naturally in heat equatio n proofs of the Index Theorem. In 2,3, we discuss further developments in index theory which lead to spectral invariants, the eta invariant and the determinant of an elliptic operator, that are definitely nonlo cal. Finally, in 4 we point out some recent connections among these no nlocal invariants and classical index theory.