On the relative index theorem. (Un théorème de l’indice relatif.) (French) Zbl 0919.58061

Carron, Gilles

Séminaire de théorie spectrale et géométrie. Année 1996-1997. St. Martin D’Hères: Univ. de Grenoble I, Institut Fourier, Sémin. Théor. Spectrale Géom., Chambéry-Grenoble. 15, 193-202 (1997).

A Riemannian manifold \({\mathcal M}\) with Riemannian metric \(g\) is called “non-parabolic” if the Laplace-Beltrami operator associated with \(({\mathcal M},g)\) admits a positive Green function. Starting from an equivalent characterization of non-parabolicity in terms of inequalities, the author calls a Dirac-type operator \(D:C^\infty(E)\to C^\infty(E)\), \(E\) a Hermitian fiber-bundle over \(M\), non-parabolic at infinity if there is a compact \(K\) in \(M\) so that for every relatively compact open set \(U\) in \(M\setminus K\) there is a constant \(C(U)\) with \(C(U)\| g\|_{L^2(U)}\leq\| Dg\|_{L^2(M\setminus K)}\), for all \(g\in C_0^\infty (M\setminus K,E)\).
The article is to a large extent a survey paper on index and relative-index theorems for Dirac operators which are non-parabolic at infinity.
For the entire collection see [Zbl 0882.00016].

Reviewer: O.Liess (Darmstadt)

MSC:

58J20	Index theory and related fixed-point theorems on manifolds
58J05	Elliptic equations on manifolds, general theory
47A53	(Semi-) Fredholm operators; index theories
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

survey; index; relative index; Dirac operators; non-parabolic at infinity