  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 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.
