Rigid body equations on spaces of pseudo-differential operators with renormalized trace

Abstract

We equip the regular Fr\'echet 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.