A generalization of the Gilkey-Branson-Fulling formula (Q2770373)

For a closed, \(n\)-dimensional, orientable, Riemannian manifold \(M\), the formula of \textit{P. Gilkey, Th. P. Branson} and \textit{S. A. Fulling} [J. Math. Phys. 32, No. 8, 2089-2091 (1991; Zbl 0778.58062)] is generalized in the following way: instead of using the de~Rham derivative and coderivative, they consider the differential \(d_h\) and codifferential \(\delta_h\) induced by a non-singular \((1,1)\)-tensor field \(h\) on \(M\). More precisely, let \(D_h^{(p)}=a^2 d_h\delta_h+b^2\delta_h d_h\) acting on \(p\)-forms; in particular, for \(a=b=1\), \(D_h^{(p)}\) equals the Laplacian \(\Delta_h^{(p)}\) induced by \(h\) on \(p\)-forms. As \(t\to 0^+\), there is an asymptotic expansion NEWLINE\[NEWLINE \text{ Tr}\left(e^{-D_h^{(p)}}\right) \sim(4\pi)^{n/2}\sum_{m=0}^\infty a_k\left(D_h^{(p)}\right) t^{2m-2n} NEWLINE\]NEWLINE of Minakshisundaram-Pleijel type, where each \(a_m\left(D_h^{(p)}\right)\) is a local invariant of \(D_h^{(p)}\). Then the authors show that NEWLINE\[NEWLINE a_m\left(D_h^{(p)}\right) =b^{2m-n} a_m\left(\Delta_h^{(p)}\right) +\left(b^{2m-n}-a^{2m-n}\right)\sum_{j\leq p}(-1)^{j-p} a_m\left(\Delta_h^{(j)}\right).NEWLINE\]NEWLINENEWLINENEWLINEFor the entire collection see [Zbl 0969.00064].