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

Sining Wei; Yong Wang

paper

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

arXiv:2609.08401

Abstract

Let $M^{n}$ be a closed oriented manifold of even dimension $n = 2 m \geq 2$ , equipped with a smooth metric $g_{1}$ and a Riemannian metric $g_{2}$ . We compute the Wodzicki residue of $D_{g_{1}}^{2} D_{g_{2}}^{- n}$ on the common exterior bundle, where $D_{g_{r}} = d + δ_{g_{r}}$ . This is the noncommutative integral of $D_{g_{1}}^{2}$ relative to the reference operator $D_{g_{2}}$ . Its local density involves the curvatures of the two metrics and the difference of their Levi--Civita connections. Integration by parts gives a closed-manifold formula without explicit derivatives of the connection difference. For compact manifolds with boundary and $n \geq 4$ , we assume that both metrics are Riemannian and satisfy $g_{r} = h_{r} (x_{n})^{- 1} g^{\partial M} + d x_{n}^{2}$ near the boundary, with $h_{r} (0) = 1$ . We compute the noncommutative residues of two factorizations involving even and odd powers of $D_{g_{2}}$ . Their interior contributions coincide, whereas their boundary terms are explicit multiples of the integral of $K_{g_{1}} - K_{g_{2}}$ , with the coefficient for the even factorization twice that for the odd one. Here $K_{g_{r}}$ is the trace of the second fundamental form with respect to the inward unit normal. The calculation uses a second-order residue formula and direct boundary symbol expansions. When the metrics coincide, the formulas reduce to the Hodge--de Rham Kastler--Kalau--Walze identity.