We study cohomological obstructions to the existence of global conserved quantities. In particular, we show that, if a given local variational problem is supposed to admit global solutions, certain cohomology classes cannot appear as obstructions. Vi
ce versa, we obtain a new type of cohomological obstruction to the existence of global solutions for a variational problem.
In view of Ehlers-Pirani-Schild formalism, since 1972 Weyl geometries should be considered to be the most appropriate and complete framework to represent (relativistic) gravitational fields. We shall here show that in any given Lorentzian spacetime (
M,g) that admits global timelike vector fields any such vector field u determines an essentially unique Weyl geometry ([g], Gamma) such that u is Gamma-geodesic (i.e. parallel with respect to Gamma).
We shall present here a general apt technique to induce connections along bundle reductions which is different from the standard restriction. This clarifies and generalizes the standard procedure to define Barbero-Immirzi (BI) connection, though on s
pacetime. The standard spacial BI connection used in LQG is then obtained by its spacetime version by standard restriction. The general prescription to define such a reduced connection is interesting from a mathematical viewpoint and it allows a general and direct control on transformation laws of the induced object. Moreover, unlike what happens by using standard restriction, we shall show that once a bundle reduction is given, then any connection induces a reduced connection with no constraint on the original holonomy as it happens when connections are simply restricted.
We discuss constraint structure of extended theories of gravitation (also known as f(R) theories) in the vacuum selfdual formulation introduced in ref. [1].
