The Bimetric generalization of Kastler--Kalau--Walze Type Theorems
Abstract
Let Mn be a closed oriented manifold of even dimension n=2m≥2, equipped with a smooth metric g1 and a Riemannian metric g2. We compute the Wodzicki residue of Dg12Dg2−n on the common exterior bundle, where Dgr=d+δgr. This is the noncommutative integral of Dg12 relative to the reference operator Dg2. 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≥4, we assume that both metrics are Riemannian and satisfy gr=hr(xn)−1g∂ M+dxn2 near the boundary, with hr(0)=1. We compute the noncommutative residues of two factorizations involving even and odd powers of Dg2. Their interior contributions coincide, whereas their boundary terms are explicit multiples of the integral of Kg1−Kg2, with the coefficient for the even factorization twice that for the odd one. Here Kgr 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.
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BibTeX
@misc{wei2026bimetric,
title = {{The Bimetric generalization of Kastler--Kalau--Walze Type Theorems}},
author = {Sining Wei and Yong Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08401},
eprint = {2609.08401},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08401},
}
RIS
TY - UNPB
TI - The Bimetric generalization of Kastler--Kalau--Walze Type Theorems
AU - Sining Wei
AU - Yong Wang
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08401
UR - https://arxiv.org/abs/2609.08401
AB - Let M^n be a closed oriented manifold of even dimension n=2m≥2, 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_1^2 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≥4, we assume that both metrics are Riemannian and satisfy g_r=h_r(x_n)^−1g^∂ M+dx_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.
ER -