  This work shows the existence of a d>2 dimensional covariant "Beltrami
vielbein" that generalizes the d=2 situation. Its definition relies on a
sub-foliation \Sigma^{ADM}_{d-1}=\Sigma_{d-3}\times\Sigma_2 of the
Arnowit--Deser--Misner leaves of d-dimensional Lorentzian manifolds {\cal M}_d.
\Sigma_2 stands for the sub-foliating randomly varying Riemann surfaces in
{\cal M}_d. The "Beltrami d -bein" associated to any given generic vielbein of
{\cal M}_d is systematically determined by a covariant gauge fixing of the
Lorentz~symmetry of the latter. It is parametrized by \frac{d(d+1)}2
independent fields belonging to different categories. Each one has a specific
interpretation. The Weyl invariant field sector of the Beltrami d-bein selects
the \frac{d(d-3)}{2} physical local degrees of freedom of d>2 dimensional
gravity. The components of the Beltrami d-bein are in a one to one
correspondence with those of the associated Beltrami d-dimensional metric. The
Beltrami parametrization of the Spin connection and of the Einstein action
delivers interesting expressions. Its use might easier the search of new Ricci
flat solutions classified by the genus of the sub-manifold \Sigma_2. A
gravitational "physical gauge" choice is introduced that takes advantage of the
geometrical specificities of the Beltrami parametrization. Further restrictions
simplify the expression of the Beltrami vielbein when {\cal M}_d has a given
spatial holonomy. This point is exemplified in the case of d=8 Lorentzian
spaces with G_2\subset SO(1,7) holonomy. The Lorentzian results presented in
this paper can be extended to the Euclidean case.

High Energy Physics - Theory