  We present a theoretical study of the bandstructure and Landau levels in
bilayer graphene at low energies in the presence of a transverse magnetic field
and Rashba spin-orbit interaction in the regime of negligible trigonal
distortion. Within an effective low energy approach (L\"owdin partitioning
theory) we derive an effective Hamiltonian for bilayer graphene that
incorporates the influence of the Zeeman effect, the Rashba spin-orbit
interaction, and inclusively, the role of the intrinsic spin-orbit interaction
on the same footing. Particular attention is spent to the energy spectrum and
Landau levels. Our modeling unveil the strong influence of the Rashba coupling
$\lambda_R $ in the spin-splitting of the electron and hole bands. Graphene
bilayers with weak Rashba spin-orbit interaction show a spin-splitting linear
in momentum and proportional to $\lambda_R $, but scales inversely proportional
to the interlayer hopping energy $\gamma_1$. However, at robust spin-orbit
coupling $\lambda_R $ the energy spectrum shows a strong warping behavior near
the Dirac points. We find the bias-induced gap in bilayer graphene to be
decreasing with increasing Rashba coupling, a behavior resembling a topological
insulator transition. We further predict an unexpected assymetric
spin-splitting and crossings of the Landau levels due to the interplay between
the Rashba interaction and the external bias voltage. Our results are of
relevance for interpreting magnetotransport and infrared cyclotron resonance
measurements, including also situations of comparatively weak spin-orbit
coupling.
