Black Strings, Black Rings and State-space Manifold

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State-space geometry is considered, for diverse three and four parameter non-spherical horizon rotating black brane configurations, in string theory and M-theory. We have explicitly examined the case of unit Kaluza-Klein momentum D₁D₅P black strings, circular strings, small black rings and black supertubes. An investigation of the state-space pair correlation functions shows that there exist two classes of brane statistical configurations, viz., the first category divulges a degenerate intrinsic equilibrium basis, while the second yields a non-degenerate, curved, intrinsic Riemannian geometry. Specifically, the solutions with finitely many branes expose that the two charged rotating D₁D₅ black strings and three charged rotating small black rings consort real degenerate state-space manifolds. Interestingly, arbitrary valued M₅-dipole charged rotating circular strings and Maldacena Strominger Witten black rings exhibit non-degenerate, positively curved, comprehensively regular state-space configurations. Furthermore, the state-space geometry of single bubbled rings admits a well-defined, positive definite, everywhere regular and curved intrinsic Riemannian manifold; except for the two finite values of conserved electric charge. We also discuss the implication and potential significance of this work for the physics of black holes in string theory.