  We study how the electronic structure of the bilayer graphene (BLG) is
changed by electric field and strain from {\it ab initio} density-functional
calculations using the LMTO and the LAPW methods. Both hexagonal and Bernal
stacked structures are considered. The BLG is a zero-gap semiconductor like the
isolated layer of graphene. We find that while strain alone does not produce a
gap in the BLG, an electric field does so in the Bernal structure but not in
the hexagonal structure. The topology of the bands leads to Dirac circles with
linear dispersion in the case of the hexagonally stacked BLG due to the
interpenetration of the Dirac cones, while for the Bernal stacking, the
dispersion is quadratic. The size of the Dirac circle increases with the
applied electric field, leading to an interesting way of controlling the Fermi
surface. The external electric field is screened due to polarization charges
between the layers, leading to a reduced size of the band gap and the Dirac
circle. The screening is substantial in both cases and diverges for the Bernal
structure for small fields as has been noted by earlier authors. As a biproduct
of this work, we present the tight-binding parameters for the free-standing
single layer graphene as obtained by fitting to the density-functional bands,
both with and without the slope constraint for the Dirac cone.
