XXVII Workshop on Geometric Methods in Physics 29.06-05.07.2008

Jacob George

Projective Connections and the Algebra of Densities

A projective connection in a vector bundle should be thought of as an Ehresmann connection with the property that parallel transport induces a projective map between fibres. We give the relations between this natural notion and the Thomas-Weyl notion of 'projective equivalence classes' of linear connections. Densities on a manifold are formal objects that generalise functions and volume forms. They form a unital, commutative associative algebra which is naturally endowed with an invariant scalar product. They are equivalently understood as a subalgebra of the algebra of functions on a closely related manifold. Here we present some new relations between the set of projective connections on a manifold and its algebra of densities.