A non-holonomic 3-web is defined by two operators and such that is a projector, is involutory, and they are connected via the relation . The so-called parallelizing connection with respect to which the 3-web distributions are parallel is defined. Some simple properties of such connections are found.
The subject of this paper is the notion of the connection in a regular Poisson manifold M, defined as a splitting of the Atiyah sequence of its Lie algebroid. In the case when the characteristic foliation F is an ℝ-Lie foliation, the fibre integral operator along the adjoint bundle is used to define the Euler class of the Poisson manifold M. When M is oriented 3-dimensional, the notion of the index of a local flat connection with singularities along a closed transversal is defined. If, additionally,...
On cotangent bundles the Liouville field, the Liouville 1-form and the canonical symplectic structure d exist. In this paper interactions between these objects and -tensor fields on cotangent bundles are studied. Properties of the connections induced by the above structures are investigated.
In this paper we consider a product preserving functor of order and a connection of order on a manifold . We introduce horizontal lifts of tensor fields and linear connections from to with respect to . Our definitions and results generalize the particular cases of the tangent bundle and the tangent bundle of higher order.
Geometric constructions of connections on the higher order principal prolongations of a principal bundle are considered. Moreover, the existing differences among connections on non-holonomic, semiholonomic and holonomic principal prolongations are discussed.
We discuss the problem of prescribing the curvature of a connection on a principal bundle whose base manifold is three-dimensional. In particular, we consider the local question: Given a curvature form , when does there exist locally a connection such that is the curvature of ? When the structure group of the bundle is semisimple, a finite number of nonlinear identities arise as necessary conditions for local solvability of the curvature equation. We conjecture that these conditions are also...