  A general investigation is made into the intrinsic Riemannian geometry for
complex systems, from the perspective of statistical mechanics. The entropic
formulation of statistical mechanics is the ingredient which enables a
connection between statistical mechanics and the corresponding Riemannian
geometry. The form of the entropy used commonly is the Shannon entropy.
However, for modelling complex systems, it is often useful to make use of
entropies such as the R\'{e}nyi and Tsallis entropies. We consider, here,
Shannon, R\'{e}nyi, Tsallis, Abe and structural entropies, for our analysis. We
focus on one, two and three particle thermally excited configurations. We find
that statistical pair correlation functions, determined by the components of
the covariant metric tensor of the underlying thermodynamic geometry,
associated with the various entropies have well defined, definite expressions,
which may be extended for arbitrary finite particle systems. In all cases, we
find a non-degenerate intrinsic Riemannian manifold. In particular, any finite
particle system described in terms of R\'{e}nyi, Tsallis, Abe and structural
entropies, always corresponds to an interacting statistical system, thereby
highlighting their importance in the study of complex systems. On the other
hand, a statistical description by the Gibbs-Shannon entropy corresponds to a
non-interacting system.
