  Flux periodicity of conducting electrons on a closed surface with genus two
$g=2$ (double torus) are investigated theoretically. We examine flux
periodicity of the ground-state energy and of the wave functions as a function
of applied magnetic field. A fundamental flux period of the ground-state energy
is twice a fundamental unit of magnetic flux for uniformly applied magnetic
field, which is shown to be valid for a simple ladder geometry and carbon
double torus. Flux periodicity of the wave functions in a double torus is
complicate as compared with a simple torus ($g=1$), and an adiabatic addition
of magnetic fluxes does not provide a good quantum number for the energy
eigenstates. The results are extended to higher genus materials and the
implications of the results are discussed.
