Quotient p-Schatten metrics on spheres

Abstract

Let S(H) be the unit sphere of a Hilbert space H and Up(H) thegroup of unitary operators in H such that u−1 belongs to the p-Schatten idealBp(H). This group acts smoothly and transitively in S(H) and endows it witha natural Finsler metric induced by the p-norm kzkp = tr(zz∗)p/21/p. Thismetric is given bykvkx,p = min{kz − ykp : y ∈ gx},where z ∈ Bp(H)ah satisfies that (dπx)1(z) = z · x = v and gx denotes theLie algebra of the subgroup of unitaries which fix x. We call z a lifting of v.A lifting z0 is called a minimal lifting if additionally kvkx,p = kz0kp. Inthis paper we show properties of minimal liftings and we treat the problemof finding short curves α such that α(0) = x and ˙α(0) = v with x ∈ S(H)and v ∈ TxS(H) given. Also we consider the problem of finding short curveswhich join two given endpoints x, y ∈ S(H).