Sur les corrections de la g\'eom\'etrie thermodynamique des trous noirs
  We study thermodynamic geometry of certain black holes and black branes with and without generalized uncertainty principle or stringy $ \alpha^{\prime} $-corrections to the entropy. From this perspective, we analyze Ruppenier geometry of Reissner-Nordstr\"om black holes and show that it is well defined and corresponds to a non-interacting statistical system. We investigate that the Weinhold geometry of dilatonic black holes is regular everywhere and that of large mass Reissner-Nordstr\"om black holes in the Poincar\'e patch of $ AdS_4 $ contains certain narrow range of thermodynamically unstable regions in the statespace. We obtain that the generalized uncertainty principle corrected Ruppenier geometry of Reissner-Nordstr\"om black holes correspond to a non-interacting statistical system unlike the magnetically charged black holes. We show that the stringy $ \alpha^{\prime} $-corrections do not introduce singularity in the statespace geometry of non-supersymmetric extremal black holes in $ D= 4 $. Interestingly, the degree of scalar curvature and that of the determinant of this Ruppenier geometry can be written as an integer multiple of the order of $ \alpha^{\prime} $-correction. We further show that the statespace geometry of Gauss- Bonnet corrected supersymmetric extremal black holes in $ D=4 $ as well as non-extremal $D_1D_5$ and $D_2D_6NS_5$ black branes in $ D=10 $ is regular everywhere. Furthermore, the thermodynamic geometry of four dimensional rotating Kerr-Newman extremal black holes in Einstein-Maxwell theory is everywhere ill-defined and that of the Kaluza-Klein black holes in Einstein-Maxwell theory or the one arrising from heterotic string compactification is ill-defined only at the points of the ergo-branch. 
Bhupendra Nath Tiwari