## $$\eta$$-invariants and their adiabatic limits.(English)Zbl 0671.58037

The Atiyah-Patody-Singer eta invariant is a spectral invariant which measures the symmetry of the spectrum of a self-adjoint elliptic differential operator. The eta invariant was introduced in connection with the index theorem for manifolds X with boundary $$\partial X$$. The invariant $$\eta$$ (A) is not locally computable. That is, it cannot be obtained by integrating over $$\partial X$$, any differential form which is given in local coordinates by a canonical expression derived from the symbol of A.
This article is devoted to study the limiting value of the $$\eta$$- invariant for Dirac operators, in situations in which the metric (or part of the metric) on the underlying manifold is multiplied by the factor $$\epsilon^{-1}$$ and $$\epsilon$$ $$\to 0$$. It is shown that in the presence of an additional invertibility hypothesis, the $$\eta$$-invariant approaches a limiting value which is locally computable (or partly locally computable).
Reviewer: V.Deundjak

### MSC:

 58J20 Index theory and related fixed-point theorems on manifolds 58J70 Invariance and symmetry properties for PDEs on manifolds
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