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

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].

### 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
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