Relative index theory. (English) Zbl 0762.58026

The author studies pairs of generalized Dirac operators on complete noncompact Riemannian manifolds which agree outside compact sets. By an analytic interpretation of the difference of the integrals over their local index densities, he obtains a generalization of the relative index theorem of M. Gromov and H. B. Lawson jun. [Publ. Math., Inst. Hautes Etud. Sci. 58, 295-408 (1983; Zbl 0538.53047)]. Furthermore, under some assumptions on the curvature of the manifolds, the author proves that the pair of generalized Dirac operators defines a supersymmetric scattering theory in the sense of N. V. Borisov, W. Müller and R. Schrader [Commun. Math. Phys. 114, No. 3, 475-513 (1988; Zbl 0663.58032)] and establishes a relation between the scattering index and the relative topological index.


58J20 Index theory and related fixed-point theorems on manifolds
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
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