  In this work we compare two fundamentally different approaches to the
electronic transport in deformed graphene: a) the condensed matter approach in
which current flow paths are obtained by applying the non-equilibrium Green's
function (NEGF) method to the tight-binding model with local strain, b) the
general relativistic approach in which classical trajectories of relativistic
point particles moving in a curved surface with a pseudo-magnetic field are
calculated. The connection between the two is established in the long-wave
limit via an effective Dirac Hamiltonian in curved space. Geometrical optics
approximation, applied to focused current beams, allows us to directly compare
the wave and the particle pictures. We obtain very good numerical agreement
between the quantum and the classical approaches for a fairly wide set of
parameters, improving with the increasing size of the system. The presented
method offers an enormous reduction of complexity from irregular tight-binding
Hamiltonians defined on large lattices to geometric language for curved
continuous surfaces. It facilitates a comfortable and efficient tool for
predicting electronic transport properties in graphene nanostructures with
complicated geometries. Combination of the curvature and the pseudo-magnetic
field paves the way to new interesting transport phenomena such as bending or
focusing (lensing) of currents depending on the shape of the deformation. It
can be applied in designing ultrasensitive sensors or in nanoelectronics.
