  Graphene ripples possess peculiar essential properties owing to the strong
chemical bonds, as an investigation using first principle calculations clearly
revealed. Various charge distributions, bond lengths, energy bands, and
densities of states strongly depend on the corrugation structures, ripple
curvatures and periods. Armchair ripples belonging to a zero-gap semiconductor
display split middle-energy states, while the zigzag ripples exhibit highly
anisotropic energy bands, semi-metallic behavior implicated by the destruction
of the Dirac cone, and the newly created critical points. Their density of
states exhibit many low-lying prominent peaks and can explain the experimental
measurements. There exist certain important similarities and differences
between graphene ripples and carbon nanotubes.
