  Standard purification interlaces Hermitian and Riemannian metrics on the
space of density operators with metrics and connections on the purifying
Hilbert-Schmidt space. We discuss connections and metrics which are well
adopted to purification, and present a selected set of relations between them.
A connection, as well as a metric on state space, can be obtained from a metric
on the purification space. We include a condition, with which this
correspondence becomes one-to-one. Our methods are borrowed from elementary
*-representation and fibre space theory. We lift, as an example, solutions of a
von Neumann equation, write down holonomy invariants for cyclic ones, and ``add
noise'' to a curve of pure states.
