几乎可换的Riemannian geometry：波算子
Almost commutative Riemannian geometry: wave operators
摘要：
与任何维数为$n$的（伪）黎曼流形$M$相关联的是流形上的$n+1$维非交换微分结构$(Ω^1,\extd)$，其额外维数将经典拉普拉斯算子编码为非交换“向量场”。利用经典联络、Ricci张量和Hodge Laplacian构造了$Ω^1$上的$(Ω^2,\extd)$和一个自然的非对易无挠联络$(\nabla,σ)$.我们证明了它的广义辫子$σ:Ω^1\tensΩ^1\to Ω^1\tensΩ^1$只有当原$M$是平坦的时才服从量子Yang-Baxter或辫子关系，即它们的失效由黎曼曲率决定，而$σ^2=\id$只有当$M$是爱因斯坦的时才服从量子Yang-Baxter或辫子关系.我们表明，如果$M$有一个共形的Killing矢量场$τ$，那么交叉产品代数$C(M)\rtimes_τ\R$视为一个非交换模拟$M\times\R$有一个自然的$n+2$维演算扩展$Ω^1$和自然时空拉普拉斯算子现在直接定义的额外的维度。案例$M=\R^3$恢复了Majid-Ruegg双交叉积平坦时空模型和在其可变光速产生中使用的波算符，但现在作为一般构造的一个例子。作为应用，我们构造了一个非对易Schwarzschild黑洞上的波算子，并初步研究了它的性质。看来无限的经典红移/时间膨胀因子在事件视界是有限的。
英文摘要：
Associated to any (pseudo)-Riemannian manifold $M$ of dimension $n$ is an $n+1$-dimensional noncommutative differential structure $(Ω^1,\extd)$ on the manifold, with the extra dimension encoding the classical Laplacian as a noncommutative `vector field'. We use the classical connection, Ricci tensor and Hodge Laplacian to construct $(Ω^2,\extd)$ and a natural noncommutative torsion free connection $(\nabla,σ)$ on $Ω^1$. We show that its generalised braiding $σ:Ω^1\tensΩ^1\to Ω^1\tensΩ^1$ obeys the quantum Yang-Baxter or braid relations only when the original $M$ is flat, i.e their failure is governed by the Riemann curvature, and that $σ^2=\id$ only when $M$ is Einstein. We show that if $M$ has a conformal Killing vector field $τ$ then the cross product algebra $C(M)\rtimes_τ\R$ viewed as a noncommutative analogue of $M\times\R$ has a natural $n+2$-dimensional calculus extending $Ω^1$ and a natural spacetime Laplacian now directly defined by the extra dimension. The case $M=\R^3$ recovers the Majid-Ruegg bicrossproduct flat spacetime model and the wave-operator used in its variable speed of light preduction, but now as an example of a general construction. As an application we construct the wave operator on a noncommutative Schwarzschild black hole and take a first look at its features. It appears that the infinite classical redshift/time dilation factor at the event horizon is made finite.
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