## An abstract index theorem on non-compact Riemannian manifolds.(English)Zbl 0790.58040

Let $$D = D_ + \oplus D_ -$$ be an essentially self-adjoint supersymmetric first-order elliptic differential operator defined on a $$\mathbb{Z}_ 2$$ graded Hermitian vector bundle $$S = S_ + \oplus S_ -$$ over a complete oriented Riemannian manifold $$M$$. Moreover, $$D$$ is assumed to be Fredholm which the author shows to imply (and in fact to be equivalent to) the existence of a constant $$c>0$$ and a compact set $$K \subset M$$ such that $\| Ds\| \geq c \| s\| \quad \text{for}\quad s\in C^ \infty_ 0(S)\quad \text{with}\quad \text{supp}(s)\cap K = \emptyset.$ Then $L^ 2\text{-index}(D_ +) = \int_ K \text{ch}\sigma(D_ +) \wedge \tau(M) + J_{M\setminus K}$ where the integrand is the usual index form (given only by local data) and where the second contribution $$J_{M\setminus K}$$ depends only on the restriction of $$D$$ to $$M\setminus K$$.
To show that $$J_{M\setminus K}$$ does not depend on the local expression of $$D$$ on $$K$$ the author uses (and proves) a relative index theorem that generalizes 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)]. Other applications of this relative index theorem are given in the author’s paper in Geom. Funct. Anal. 3 (1993, to appear)].

### MSC:

 58J20 Index theory and related fixed-point theorems on manifolds 53C20 Global Riemannian geometry, including pinching 47A53 (Semi-) Fredholm operators; index theories 58J22 Exotic index theories on manifolds 47B25 Linear symmetric and selfadjoint operators (unbounded) 81Q60 Supersymmetry and quantum mechanics

Zbl 0538.53047