  There is convincing numerical evidence that fractional quantum Hall
(FQH)-like ground states arise in fractionally filled Chern bands (FCB). Here
we show that the Hamiltonian theory of Composite Fermions (CF) can be as useful
in describing the FCB as it was in describing the FQHE in the continuum. We are
able to introduce CFs into the FCB problem even though there is no external
magnetic field by following a two-stage process. First we construct an
algebraically exact mapping which expresses the electron density projected to
the Chern band, ${\rho}_{{\tiny FCB}}$, as a sum of Girvin-MacDonald-Platzman
density operators, ${\rho}_{{\tiny GMP}}$, that obey the Magnetic Translation
Algebra. Next, following our Hamiltonian treatment of the FQH problem, we
rewrite the GMP operators in terms of CF variables which reproduce the same
algebra. This naturally produces a unique Hartree-Fock ground state for the
CFs, which can be used as a springboard for computing gaps, response functions,
temperature-dependent phenomena, and the influence of disorder. We give two
concrete examples, one of which has no analog in the continuum FQHE with $\nu=
{1 \over 5}$ and $\sigma_{xy}={2\over 5}$. Our approach can be easily extended
to fractionally filled, strongly interacting two-dimensional
time-reversal-invariant topological insulators.
