具有重整化迹的伪微分算子空间上的刚体方程
Rigid body equations on spaces of pseudo-differential operators with renormalized trace
摘要：
我们装备的正规弗雷歇李群可逆的，奇数类，经典的pseudodiomatic运营商$Cl^{0,*}_{odd}(M,E)$ -其中$M$是一个紧凑的光滑流形和$E$（复杂）向量丛上$M$-伪黎曼度量，我们使用这些度量引入了一大类刚体方程。我们适应我们的无穷维设置Manakov的经典观察的可积性欧拉方程的刚体，我们表明，我们的方程可以写在拉克斯形式（参数），他们承认一个无限数量的运动积分。我们还证明了度量连接的存在性，我们表明，我们的刚体方程确定测地线上$Cl^{0,*}_{odd}(M,E)$，我们提出了严格的公式，相应的曲率和截面曲率。我们的主要工具是无边界紧致光滑流形上伪微分算子的重整化迹理论。
英文摘要：
We equip the regular Fréchet Lie group of invertible, odd-class, classical pseudodifferential operators $Cl^{0,*}_{odd}(M,E)$ -- in which $M$ is a compact smooth manifold and $E$ a (complex) vector bundle over $M$ -- with pseudo-Riemannian metrics, and we use these metrics to introduce a large class of rigid body equations. We adapt to our infinite-dimensional setting Manakov's classical observation on the integrability of Euler's equations for the rigid body, and we show that our equations can be written in Lax form (with parameter) and that they admit an infinite number of integrals of motion. We also prove the existence of metric connections, we show that our rigid body equations determine geodesics on $Cl^{0,*}_{odd}(M,E)$, and we present rigorous formulas for the corresponding curvature and sectional curvature. Our main tool is the theory of renormalized traces of pseudodifferential operators on compact smooth manifolds without boundary.
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