Let M be a finite von Neumann algebra with a faithful trace τ . In this paper we study metric geometry of homogeneous spaces O of the unitary group UM of M, endowed with a Finsler quotient metric induced by the p-norms of τ , ‖x‖p = τ(|x|p)1/p, p ≥ 1. The main results include the following. The unitary group carries on a rectifiable distance dp induced by measuring the length of curves with the p-norm. If we identify O as a quotient of groups, then there is a natural quotient distance ḋp that metrizes the quotient topology. On the other hand, the Finsler quotient metric defined in O provides a way to measure curves, and therefore, there is an associated rectifiable distance dO,p . For p ≥ 2, we prove that the distances ḋp and dO,p coincide. Based on this fact, we show that the metric space (O, ḋp) is a complete path metric space. The other problem treated in this article is the existence of metric geodesics, or curves of minimal length, in O. We give two abstract partial results in this direction. The first concerns the initial values problem and the second the fixed endpoints problem. We show how these results apply to several examples. In the process, we improve some results about the metric geometry of UM with the p-norm.