  We present the electronic properties of massless Dirac fermions characterized
by geometry and topology on a graphene sheet in this chapter. Topological
effects can be elegantly illuminated by the Atiyah-Singer index theorem. It
leads to a topological invariant under deformations on the Dirac operator and
plays an essential role in formulating supersymmetric quantum mechanics over
twisted Dolbeault complex caused by the topological deformation of the lattice
in a graphene system. Making use of the G index theorem and a high degree of
symmetry, we study deformed energy eigenvalues in graphene. The Dirac fermion
results in SU(4) symmetry as a high degree of symmetry in the noninteracting
Hamiltonian of the monolayer graphene. Under the topological deformation the
zero-energy states emerge naturally without the Zeeman splitting at the Fermi
points in the graphene sheet. In the case of nonzero energy, the up-spin and
down-spin states have the exact high symmetries of spin, forming the pseudospin
singlet pairing. We describe the peculiar and unconventional quantum Hall
effects of the n = 0 Landau level in monolayer graphene on the basis of the G
index theorem and the high degree of symmetry.
