发表机构
School of Data Science and Artificial Intelligence, Dongbei University of Finance and Economics; School of Mathematics and Statistics, Northeast Normal University（东北财经大学数据科学与人工智能学院; 东北师范大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文推广Kastler--Kalau--Walze定理至双度量情形，计算Wodzicki留数，得到边界项为曲率差积分，度量重合时还原经典恒等式。
AI 中文摘要
设$M^n$为偶数维$n=2m\ge2$的闭定向流形，配备光滑度量$g_1$与黎曼度量$g_2$。我们在公共外微分丛上计算$D_{g_1}^2D_{g_2}^{-n}$的Wodzicki留数，其中$D_{g_r}=d+\delta_{g_r}$。这是$D_{g_1}^2$相对于参考算子$D_{g_2}$的非交换积分。其局部密度涉及两个度量的曲率以及它们Levi--Civita联络之差。分部积分给出闭流形公式，不显含联络差的导数。对于带边紧流形且$n\ge4$，我们假设两个度量均为黎曼度量，并在边界附近满足$g_r=h_r(x_n)^{-1}g^{\partial M}+dx_n^2$，其中$h_r(0)=1$。我们计算涉及$D_{g_2}$偶次幂和奇次幂的两个分解的非交换留数。它们的内部贡献一致，而边界项是$K_{g_1}-K_{g_2}$积分的显式倍数，其中偶次幂分解的系数是奇次幂分解的两倍。这里$K_{g_r}$是相对于内向单位法向的第二基本形式的迹。计算使用二阶留数公式和直接边界符号展开。当度量重合时，公式退化为Hodge--de Rham Kastler--Kalau--Walze恒等式。
英文摘要
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.
