The Bimetric generalization of Kastler--Kalau--Walze Type Theorems

Let $M^n$ be a closed oriented manifold of even dimension $n=2m\ge2$, equipped with a smooth metric $g_1$ and a Riemannian metric $g_2$. We compute the Wodzicki residue of $D_{g_1}^2D_{g_2}^{-n}$ on the common exterior bundle, where $D_{g_r}=d+δ_{g_r}$. This is the noncommutative integral of $D_{g_1