For noncompact hypersurfaces satisfying mild geometric conditions, the partitioned index of a Dirac operator equals the index of the induced Dirac operator on the hypersurface.
Conway.A course in functional analysis, volume 96 ofGraduate Texts in Mathematics
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A partitioned manifold index theorem for noncompact hypersurfaces
For noncompact hypersurfaces satisfying mild geometric conditions, the partitioned index of a Dirac operator equals the index of the induced Dirac operator on the hypersurface.