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On the $$L^ 2$$-index of Dirac operators on manifolds with corners of codimension two. I. (English) Zbl 0881.58071
The purpose of this paper is to generalize the Atiyah-Patodi-Singer index theorem to compact manifolds with corners. A manifold with corners can be defined as a topological manifold $$X$$ with boundary together with an embedding $$\iota: X\hookrightarrow \widetilde X$$ into a closed $$C^\infty$$ manifold for which there exists a finite collection of functions $$\rho_i\in C^\infty (\widetilde X)$$, $$i\in I$$, such that $$\iota(X) =\{x\in \widetilde X\mid \rho_i (x)\geq 0$$, $$i\in I\}$$ and for each subset $$J\subset I$$, the $$d\rho_i$$, $$i\in J$$, are linearly independent at each point $$x\in\widetilde X$$ where all $$\rho_i$$, $$i\in J$$, vanish. The boundary of $$X$$ is the union of embedded hypersurfaces $$Y_i$$, $$i\in I$$.
Let $$Y_{i_1 \dots i_k}= Y_{i_1} \cap \cdots \cap Y_{i_k}$$, $$i_j\in I$$. Then we say that $$Y_{i_1 \dots i_k}$$ is a corner of codimension $$k$$. Let $$X$$ be endowed with a metric which is a product near all hypersurfaces and also near all corners. Let a Dirac operator on $$X$$ be adapted to the product structure near the boundary. Then the goal is to generalize the Atiyah-Patodi-Singer index theorem to this case. In this paper only manifolds with corners of codimension $$\leq 2$$ are considered.

##### MSC:
 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 58J20 Index theory and related fixed-point theorems on manifolds
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