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A general investigation is made into the intrinsic Riemannian geometry forcomplex systems, from the perspective of statistical mechanics. The entropicformulation of statistical mechanics is the ingredient which enables aconnection between statistical mechanics and the corresponding Riemanniangeometry. The form of the entropy used commonly is the Shannon entropy.However, for modelling complex systems, it is often useful to make use ofentropies such as the R\'{e}nyi and Tsallis entropies. We consider, here,Shannon, R\'{e}nyi, Tsallis, Abe and structural entropies, for our analysis. Wefocus on one, two and three particle thermally excited configurations. We findthat statistical pair correlation functions, determined by the components ofthe 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, wefind a non-degenerate intrinsic Riemannian manifold. In particular, any finiteparticle system described in terms of R\'{e}nyi, Tsallis, Abe and structuralentropies, always corresponds to an interacting statistical system, therebyhighlighting their importance in the study of complex systems. On the otherhand, a statistical description by the Gibbs-Shannon entropy corresponds to anon-interacting system.

A thermodynamic geometric study of R\'{e}nyi and Tsallis entropies