Heat trace asymptotics and the Gauss-Bonnet theorem for general connections (Q2920318)

Let \(M\) be a closed Riemannian \(m\)-dimensional manifold and let \(D=\begin{pmatrix} 0&D_-\\D_+&0\end{pmatrix}\) be a Dirac like operator on the sections to a \(\mathbb Z_2\)-graded vector bundle \(E=E_+\oplus E_-\) on \(M\). For geometric operators \(D\) like the signature operator, the heat operators \(e^{-tD_+D_-},e^{-tD_-D_+}\) are of trace class for \(t>0\) and the trace difference is \(t\)-independent, with value \(\tau_D\) given by the integral on \(M\) of a local density \(e_D(t,x)\) with an asymptotic expansion in small \(t\) : \(e_D(t,x)\sim_{t\to0^+}\sum_{n\geq0}t^{(n-m)/2}e_{D,n}(x)\) with \(\int_Me_{D,n}(x)=0\) if \(n<m\) and \(\tau_D=\int_Me_{D,m}(x)\).NEWLINENEWLINEFor the de Rham complex \((E=\bigwedge T^*M,D=d+d^*)\), \textit{V. K. Patodi} [J. Differ. Geom. 5, 233--249 (1971; Zbl 0211.53901)] proved cancellation of all the local densities \(e_{D,n}(x)\), \(n<m\) and constancy of \(e_{D,m}(x)\). The paper handles operators \(D\) induced by super-connections \(\nabla\) on the fiber bundle \(E=\bigwedge T^*M\) : the preceding spectral properties (e. g. trace class heat operators, with trace difference \(\tau_D\) given by the integration of a density) remain and you can wonder if the Patodi cancellations persist. If the connection \(\nabla\) obeys the Leibniz rule, the authors prove the cancellation \(e_{D,n}(x)=0\) for \(3n<2m\). Moreover, they construct a connection on \(\bigwedge^*\mathbb T^m\) with non zero local densities \(e_{D,{2n}}(x_0)\) for every \(n \geq1\) and given \(x_0\in\mathbb T^m\).