  The geometry of two-dimensional crystalline membranes dictates their
mechanical, electronic and chemical properties. The local geometry of a surface
is determined from the two invariants of the metric and the curvature tensors.
Here we discuss those invariants directly from atomic positions in terms of
angles, areas, vertex and normal vectors from carbon atoms on the graphene
lattice, for arbitrary elastic regimes and atomic conformations, and without
recourse to an effective continuum model. The geometrical analysis of graphene
membranes under mechanical load is complemented with a study of the local
density of states (LDOS), discrete induced gauge potentials, velocity
renormalization, and non-trivial electronic effects originating from the scalar
deformation potential. The asymmetric LDOS is related to sublattice-specific
deformation potential differences, giving rise to the pseudomagnetic field. The
results here enable the study of geometrical, mechanical and electronic
properties for arbitrarily-shaped graphene membranes in experimentally-relevant
regimes without recourse to differential geometry and continuum elasticity.
