Hermitian Structures and Compatible Connections on A-bundles
Abstract
Description
A-manifolds and A-bundles are manifolds and vector bundles modelled on a projective finitely generated module over a topological algebra A. In this paper we investigate the conditions under which an A-bundle is provided with an A-valued hermitian structure and a compatible connection, in case A is a commutative complete locally m-convex C*-algebra with unit. In this investigation two obstacles appear: first, A-manifolds do not admit partitions of unity in the classical sence, and, secondly, the existence of a hermitian structure is not equivalent to the reduction of the structural group of the bundle to a certain subgroup. However, we prove that if the bundle has a trivializing covering whose transition functions take values in the group of the "A-hermitian product preserving" automorphisms of the fibre type and the base space admits at least one A-valued partition of unity (subordinate to this covering), then the A-bundle admits an A-hermitian structure and a compatible connection.
Latex, 12 pages, to be published in the Proc. 2nd Conf. of Balkan Soc of Geometers, Thessaloniki, 1998
Latex, 12 pages, to be published in the Proc. 2nd Conf. of Balkan Soc of Geometers, Thessaloniki, 1998