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 ${\alpha }^{\mathrm{\prime }}$-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 $Ad{S}_4$ 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 ${\alpha }^{\mathrm{\prime }}$-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 ${\alpha }^{\mathrm{\prime }}$-correction. We further show that the statespace geometry of Gauss- Bonnet corrected supersymmetric extremal black holes in $D=4$ as well as non-extremal $D_1{D}_5$ and $D_2{D}_{6}N{S}_5$ 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.