  A single two-dimensinoal Dirac fermion state has been recently observed on
the surface of topological insulator Bi$_2$Te$_3$ by angle-resolved
photoemission spectroscopy (ARPES). We study the surface band structure using
$k \cdot p$ theory and find an unconventional hexagonal warping term, which is
the counterpart of cubic Dresselhaus spin-orbit coupling in rhombohedral
structures. We show that this hexagonal warping term naturally explains the
observed hexagonal snow-flake Fermi surface. The strength of hexagonal warping
is characterized by a single parameter, which is extracted from the size of the
Fermi surface. We predict a number of testable signatures of hexagonal warping
in spectroscopy experiments on Bi$_2$Te$_3$. We also explore the possibility of
a spin-density wave due to strong nesting of the Fermi surface.
