  The inner geometry induced by the Bures distance function on the state space
of a unital C*-algebra, or on parts of it, is considered in detail. As a key
result the local dilation function is calculated at a state when the state
argument is bound to vary along certain parameterized curves passing through
this state. The parameterized curves considered are those possessing
differentiable local implementations as vector states relative to some unital
*-representation in the vicinity of the state in question. The space of all
tangent forms at a given state correspondent to this category of curves is
specified as normed linear space with a quadratic norm which is depending from
the state in a characteristic manner. In analyzing the structure of the local
dilation function special emphasis is laid on describing such parts of the
state space, in restriction to which the line element would be of Finslerian
type, and which were geodesically convex subsets. All results obtained are
illustrated by examples, and the implications obtained are compared to those
known from the investigations of Uhlmann et al in the finite dimensional case.
