Topological phases of matter have revolutionized quantum engineering. Implementing a curved space Dirac equation solver based on the quantum Lattice Boltzmann method, we study the topological and geometrical transport properties of a Mobius graphene ribbon. In the absence of a magnetic field, we measure a quantum spin-Hall current on the graphene strip, originating from topology and curvature, whereas a quantum Hall current is not observed. In the torus geometry a Hall current is measured. Additionally, a specific illustration of the equivalence between the Berry and Ricci curvature is presented through a travelling wave-packet around the Mobius band.
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