Heat equation approach to index theorems on odd dimensional manifolds (Q1043701)

This paper studies the index of the Dirac operator on odd-dimensional manifolds with boundary. Let \(M\) be a compact spin manifold of dimension \(2n+1\) with boundary \(\partial M\) with a Riemannian metric taking the product form \(du^2+g\) in a collar neighborhhod \(U\) of the boundary.Let \(S\) be a spin bundle over \(M\). Then the restriction of \(S\) to the collar neighborhood splits into the positive and negative parts \(S_{|U}=S^+\oplus S^-\) of the spin bundle. Let \(N_i\) be the connected components of the boundary and \(\varepsilon_i\) be \(0\), \(+\) or \(-\) and be fixed arbitrarily for each \(N_i\). Let \(A_i\) be the Dirac operators on \(N_i\). Let \(P^\pm\) denote the boundary conditions defined by \((\phi_{|N_i})^\pm=0\) and \(P^0\) denote the Atiyah-Patodi-Singer boundary conditions, that is, \(P^0\) is the orthogonal projection on the positive part of the spectrum of the Dirac operator on the boundary. Let \(P^\varepsilon\) denote the set of all boundary conditions. Then the index of the elliptic boundary value problem \((D, P^\varepsilon)\) is defined by \[ \text{ind}\,(D,P^\varepsilon)=\text{dim}\,\text{ker}\,D^*D -\text{dim}\,\text{ker}\,DD^* \] By using the heat kernel methods the author proves the following theorem \[ \text{ind}(D,P^\varepsilon) =\frac{1}{2}\sum_{\varepsilon_i=-}\text{ind}\,A_i -\frac{1}{2}\sum_{\varepsilon_i=+}\text{ind}\,A_i -\frac{1}{2}\sum_{\varepsilon_i=0}\text{dim}\,\text{ker}\,A_i \] As a consequence the author also obtains some information about the isospectral invariants of the boundary conditions as well as generalizes this theorem to a family of Dirac operators.