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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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