Classical A_n--W-Geometry

This is a detailed development for the $A_n$ case, of our previous article entitled "W-Geometries" to be published in Phys. Lett. It is shown that the $A_n$--W-geometry corresponds to chiral surfaces in $CP^n$. This is comes out by discussing 1) the extrinsic geometries of chiral surfaces (Frenet-Serret and Gauss-Codazzi equations) 2) the KP coordinates (W-parametrizations) of the target-manifold, and their fermionic (tau-function) description, 3) the intrinsic geometries of the associated chiral surfaces in the Grassmannians, and the associated higher instanton- numbers of W-surfaces. For regular points, the Frenet-Serret equations for $CP^n$--W-surfaces are shown to give the geometrical meaning of the $A_n$-Toda Lax pair, and of the conformally-reduced WZNW models, and Drinfeld-Sokolov equations. KP coordinates are used to show that W-transformations may be extended as particular diffeomorphisms of the target-space. This leads to higher-dimensional generalizations of the WZNW and DS equations. These are related with the Zakharov- Shabat equations. For singular points, global Pl\"ucker formulae are derived by combining the $A_n$-Toda equations with the Gauss-Bonnet theorem written for each of the associated surfaces.Comment: (60 pages

The York map as a Shanmugadhasan canonical transformation in tetrad gravity and the role of non-inertial frames in the geometrical view of the gravitational field

A new parametrization of the 3-metric allows to find explicitly a York map in canonical ADM tetrad gravity, the two pairs of physical tidal degrees of freedom and 14 gauge variables. These gauge quantities (generalized inertial effects) are all configurational except the trace ${}^3K(\tau ,\vec \sigma)$ of the extrinsic curvature of the instantaneous 3-spaces $\Sigma_{\tau}$ (clock synchronization convention) of a non-inertial frame. The Dirac hamiltonian is the sum of the weak ADM energy $E_{ADM} = \int d^3\sigma {\cal E}_{ADM}(\tau ,\vec \sigma)$ (whose density is coordinate-dependent due to the inertial potentials) and of the first-class constraints. Then: i) The explicit form of the Hamilton equations for the two tidal degrees of freedom in an arbitrary gauge: a deterministic evolution can be defined only in a completely fixed gauge, i.e. in a non-inertial frame with its pattern of inertial forces. ii) A general solution of the super-momentum constraints, which shows the existence of a generalized Gribov ambiguity associated to the 3-diffeomorphism gauge group. It influences: a) the explicit form of the weak ADM energy and of the super-momentum constraint; b) the determination of the shift functions and then of the lapse one. iii) The dependence of the Hamilton equations for the two pairs of dynamical gravitational degrees of freedom (the generalized tidal effects) and for the matter, written in a completely fixed 3-orthogonal Schwinger time gauge, upon the gauge variable ${}^3K(\tau ,\vec \sigma)$, determining the convention of clock synchronization. Therefore it should be possible (for instance in the weak field limit but with relativistic motion) to try to check whether in Einstein's theory the {\it dark matter} is a gauge relativistic inertial effect induced by ${}^3K(\tau ,\vec \sigma)$.Comment: 90 page