Riemann scalar curvature of ideal quantum gases obeying Gentile's statistics

The scaler curvature (R) of ideal quantum gases obeying Gentile's statistics is investigated by the method of information geometrical theory. The R value is specified by the fugacity eta and the maximum number, p, of particles in a state. The lowest case p = 1, corresponds to Fermi-Dirac statistics and the unbounded case, p --> infinity, to Bose-Einstein statistics. In contrast to R = 0 for ideal classical gases obeying Boltzmann statistics, we find R = root 2/32 for p greater than or equal to 2 and R = -2 root 2/32 for p = 1, in eta --> 0 which is the classical limit. This means that a quantum statistical character is left in R, in the classical limit. Also, a correlation between the sign of R and a quantum mechanical exchange effect is recognized for eta --> 0 and eta >> 1. Furthermore, we obtain results that support the instability interpretation of R proposed by Janyszek and Mrugala.