  We present an argument justifying the origin of the escort distributions used
in calculations involving the Tsallis entropy. We rely on an induced hyperbolic
Riemannian metric reflecting the generalized composition property of the
Tsallis entropy. The mapping of the corresponding Riemannian measure on the
space of thermodynamic variables gives the specific form of the escort
distributions and provides a geometric interpretation of the non-extensive
parameter. In addition, we explain the polynomial rate of increase of the
sample space volume for systems described by the Tsallis entropy, thus
extending the previously reached conclusions for discrete systems to the case
of systems whose evolution is described by flows on Riemannian manifolds.
