{
  "abstract": "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 $D_{g_1}^2$ 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 $n\\ge4$, we assume that both metrics are Riemannian and satisfy $g_r=h_r(x_n)^{-1}g^{\\partial M}+dx_n^2$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08401",
  "author_total": 2,
  "authors": [
    {
      "name": "Sining Wei"
    },
    {
      "name": "Yong Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fyk78hm84zk3g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08401",
  "fulltext_url": null,
  "id": "fyk78hm84zk3g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08401",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7212000991",
  "pdf_url": "https://arxiv.org/pdf/2609.08401",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fyk78hm84zk3g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08401",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08401"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The Bimetric generalization of Kastler--Kalau--Walze Type Theorems",
  "title_text": "The Bimetric generalization of Kastler--Kalau--Walze Type Theorems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/fyk78hm84zk3g",
  "venue": "arXiv",
  "versions": []
}