  We consider the Hamiltonian formulation of Horava gravity in arbitrary
dimensions, which has been proposed as a renormalizable gravity model for
quantum gravity without the ghost problem. We study the "full" constraint
analysis of the "non-projectable" Horava gravity whose potential, V(R), is an
arbitrary function of the (intrinsic) Ricci scalar R. We find that there exist
generally distinct cases of this theory, depending on (i) whether the
Hamiltonian constraint generates new (second-class) constraints (Cases A, C) or
just fixes the associated Lagrange multipliers (Case B), or (ii) whether the IR
Lorentz-deformation parameter \lambda is at the conformal point (Case C) or not
(Cases A, B). It is found that, for Cases A and C, the dynamical degrees of
freedom are the same as in general relativity, while, for Case B, there is "one
additional phase-space degree of freedom", representing an extra (odd) scalar
graviton mode. This would resolve the long-standing debates about the extra
graviton modes and achieves the dynamical consistency of the Horava gravity, at
the "fully non-linear" level. Several exact solutions are also considered as
some explicit examples of the new constraints. The structure of the newly
obtained, "extended" constraint algebra seems to be generic to Horava gravity
and its general proof would be a challenging problem. Some other challenging
problems, which include the path integral quantization and the Dirac bracket
quantization are discussed also.
