  We discuss the effect of elastic deformations on the electronic properties of
bilayer graphene membranes. Distortions of the lattice translate into
fictitious gauge fields in the electronic Dirac Hamiltonian which are
explicitly derived here for arbitrary elastic deformations. We include gauge
fields associated to intra- as well as inter-layer hopping terms and discuss
their effects in different contexts. As a first application, we use the gauge
fields in order to study the recently predicted strain-induced Lifshitz
transition for the Fermi surface at low energy. As a second application, we
discuss the electron-phonon coupling induced by the fictitious gauge fields and
analyse its contribution to the electrical resistivity of suspended bilayer
membranes. Of special interest is the appearance of a linear coupling for
flexural modes, in stark contrast to the case of monolayer graphene. This new
coupling channel is shown to dominate the temperature-dependent resistivity in
suspended samples with low tension.
