RIGID BODY EQUATIONS ON SPACES OF PSEUDO-DIFFERENTIAL OPERATORS WITH RENORMALIZED TRACE

Jean-Pierre Magnot (1) , Enrique G Reyes

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

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 class of rigid body equations. We prove the existence of a metric connection, 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.

Mots clés: rigid body equations; pseudodifferential operators; renormalized traces

Domaines: Géométrie différentielle [math.DG]; Physique mathématique [math-ph]; Algèbres d'opérateurs [math.OA]; Théorie spectrale [math.SP]; Systèmes Solubles et Intégrables [nlin.SI]

Jean-Pierre Magnot, Enrique G Reyes. RIGID BODY EQUATIONS ON SPACES OF PSEUDO-DIFFERENTIAL OPERATORS WITH RENORMALIZED TRACE. 2021. ⟨hal-03194881⟩