Efstathios Vassiliou

Transformations of linear connections, II

We elucidate [9] with two applications. In the first we view connections as differential systems. Specializing this to trivial bundles overS1 and applying the theory of Floquet, we obtain equivalent connections with constant Christoffel symbols. In the second application we prove that the canonical connections of parallelizable manifolds (in particular Lie groups) can be obtained from the canonical flat connection of appropriate trivial bundles. Thus, the formalisms of [1], [4], [5] and [6] fit in the general setting of [9].

Isomorphism classes for Banach vector bundle structures of second tangents

On a smooth Banach manifold M, the equivalence classes of curves that agree up to acceleration form the second order tangent bundle T^2M of M. This is a vector bundle in the presence of a linear connection on M and the corresponding local structure is heavily dependent on the choice of connection. In this paper we study the extent of this dependence and we prove that it is closely related to the notions of conjugate connections and second order differentials. In particular, the vector bundle structure on T^2M remains invariant under conjugate connections with respect to diffeomorphisms of M.

Connections on Principal Sheaves

We define and study connections on principal sheaves locally isomorphic to appropriate sheaves of groups. Their relationship with connections on principal fibre bundles, connections on vector sheaves (in the sense of [13]) and the early ideas of [1] is also investigated. Only sheaf-theoretic methods are employed, without any differentiability. This enlarges the framework of the classical geometry to non-smooth spaces and might be of interest to modern physics.

(f, ϕ,h)-related connections and Liapunoff's theorem

SummaryLetl=(P, G, B, π) (resp.l′) be a principal bundle endowed with a connection ω (resp. ω′) and let (f, ϕ,h) be a morphism ofl intol′. Roughly speaking, ω and ω′ are (f, ϕ,h)-related it the morphism preserves the horizontal subspaces.The main result is a criterion for such a relationship, under aG-B-isomorphism, given in terms of the corresponding local connection forms.Since the connections on finite-dimensional trivial bundles correspond to ordinary differential systems, the above result leads to the usual transformations of (equivalent) systems and to the condition for the existence of a system with constant coefficient (Liapunoff).

Flat bundles and holonomy homomorphisms

Using the natural equivalence relation in the set of flat Banach principal fibre bundles with group G and connected base B, we obtain a bisection between the corresponding equivalence classes and classes of similar homomorphisms of π1(B) into G.

UNIVERSAL CONNECTIONS ON LIE GROUPOIDS

Given a Lie groupoid Ω , we construct a groupoid J 1 Ω equipped with a universal connection from which all the connections of Ω are obtained by certain pullbacks. We show that this general construction leads to universal connections on principal bundles (considered by Garcia (1972)) and universal linear connections on vector bundles (ultimately related with those of Cordero et al. (1989)).

On a class of principal bundles over symplectic bases on euclidean spaces

In this paper we are concerned with a certain class of principal fibre bundles over symplectic manifolds, namely bundles equipped with a connection whose curvature form is, roughly speaking, the lift of a symplectic form. This situation appears e.g. in the geometric (pre)quantization where the structural group is assumed to be abelian. However, the commutativity of the group is a simplifying assumption, which conceals a great deal of the geometric aspect underlying the amphidromy between the global and local structure of such bundles. The purpose of the present note is precisely to exhibit the previous geometric aspect and to characterize equivalent bundles of the above said type in terms of appropriate local functions and forms. This is stated in the Theorem of Section 2. We should like to add that, beside the non-commutativity of the structure group, our proof does not rely on particular open coverings (such as l-simple) of the base space, a fact which allows to set the whole approach within the framework of infinite-dimensional manifolds and bundles.