  Within the framework of geodetic brane gravity, the Universe is described as
a 4-dimensional extended object evolving geodetically in a higher dimensional
flat background. In this paper, by introducing a new pair of canonical fields
{lambda, P_{lambda}}, we derive the quadratic Hamiltonian for such a brane
Universe; the inclusion of matter then resembles minimal coupling. Second class
constraints enter the game, invoking the Dirac bracket formalism. The algebra
of the first class constraints is calculated, and the BRST generator of the
brane Universe turns out to be rank-1. At the quantum level, the road is open
for canonical and/or functional integral quantization. The main advantages of
geodetic brane gravity are: (i) It introduces an intrinsic, geometrically
originated, 'dark matter' component, (ii) It offers, owing to the Lorentzian
bulk time coordinate, a novel solution to the 'problem of time', and (iii) It
enables calculation of meaningful probabilities within quantum cosmology
without any auxiliary scalar field. Intriguingly, the general relativity limit
is associated with lambda being a vanishing (degenerate) eigenvalue.
