  The aim of the paper is to extend the notion of $\alpha$-geometry in the
classical and in the noncommutative case by introducing a more general class of
pull-back metrics and to give concrete formulas for the scalar curvature of
these Riemannian manifolds. We introduce a more general class of pull-back
metrics of the noncommutative state spaces, we pull back the Euclidean
Riemannian metric of the space of self-adjoint matrices with functions which
have an analytic extension to a neighborhood of the interval $]0,1[$ and whose
derivative are nowhere zero. We compute the scalar curvature in this setting,
and as a corollary we have the scalar curvature of the classical probability
space when it is endowed with such a general pull-back metric. In the
noncommutative setting we consider real and complex state spaces too. We give a
simplification of Gibilisco's and Isola's conjecture for the first nontrivial
classical probability space and we present the result of a numerical
computation which indicate that the conjecture may be true for the space of
real and complex qubits.
