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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

Citations:

Zbl 0538.53047
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