HERMITIAN DIFFERENTIAL GEOMETRY ON /A-BUNDLES
Abstract
IN THIS WORK, WE FIRST DEVELOP METHODS OF DIFFERENTIATION SUITABLE FOR MAPPINGS BETWEEN TOPOLOGICAL A-MODULES, WHERE A IS A COMMUTATIVE LOCALLY M-CONVEX * -ALGEBRA WITH UNIT. THIS EXTENDS WELL-KNOWN METHODS OF DIFFERENTIATION ON TOPOLOGICAL VECTOR SPACES, SUCH AS FRECHET AND HYERS DIFFERENTIATION. APPLYING THE ABOVE DIFFERENTIAL CALCULUS, WE INTRODUCE THE CONCEPT OF DIFFERENTIABLE A-MANIFOLDS, THAT IS, DIFFERENTIABLE MANIFOLDS MODELLED ON TOPOLOGICAL (FINITELY GENERATED AND PROJECTIVE) A-MODULES. WE ALSO EXAMINE BASIC PROPERTIES OF THE RESPECTIVE TANGENT SPACES, DERIVATIONS AND VECTOR FIELDS. FINALLY, WE STUDY DIFFERENTIABLE BUNDLES OF FIBRE TYPE A TOPOLOGICAL A-MODULE AND PROVE THAT SUCH BUNDLES ARE PROVIDED WITH A DIFFERENTIABLE A- HERMITIAN STRUCTURE AND A COMPATIBLECONNECTION.
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