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Inversion asymmetry in bilayer graphene can be tuned by the displacementfield. As a result, the band dispersion in biased bilayer graphene acquiresflat band regions near the Dirac points along with a non-trivial band geometry.We analyze the effect of inversion symmetry on the critical temperature andsuperfluid stiffness of the superconducting state of AB-stacked graphenebilayer and on the exciton condensate in double layers formed by two AB-stackedgraphene bilayers. The geometric superfluid stiffness in bilayer graphenesuperconductors is found to be negligible due to the small superconducting gap.Furthermore, we show that the geometric superfluid stiffness is maximized for aconstant order parameter. Therefore, it can be neglected in biased bilayergraphene superconductors with any pairing symmetry. However, the displacementfield enhances the geometric superfluid stiffness in exciton condensates. It ismost prominent at low densities and high displacement fields. A consequence ofthe geometric superfluid stiffness is a modest enhancement of theBerezinskii-Kosterlitz-Thouless transition temperature in bilayer graphene'sexciton condensate.

Effect of Inversion Asymmetry on Bilayer Graphene's Superconducting and Exciton Condensates