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We show that the modulation of the hopping amplitudes in the honeycomblattice of the monolayer graphene uniquely defines a metric which correspondsto the shape of the Dirac cone. The spin connection of this effective metricfield can be obtained from the microscopic tight-binding Hamiltonian exactly,completing the analogy between the sublattice pseudospin travelling in themonolayer graphene with ripples and strain fields, and the real spin $1/2$fermion travelling in a curved space. The effective metric as seen by thesublattice pseudospin is different from the real space metric as defined by thetwo-dimensional manifold of the graphene monolayer. All relevant terms of theeffective gauge field from the microscopic model is calculated exactly for aunimodular effective metric.

Dirac Cone Metric and the Origin of the Spin Connections in Monolayer Graphene