Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 813This is a curvature–free connection on Y . There is the 1–1 correspondencebetween the curvature–free connections and the horizontal foliations on abundle Y −→ X.Given a horizontal foliation on Y −→ X, there exists the associated atlasof bundle coordinates (x α , y i ) of Y such that (i) every leaf of this foliation islocal generated by the equations y i = const, and (ii) the transition functionsy i −→ y ′i (y j ) are independent on the coordinates x α of the base X [Kamberand Tondeur (1975)]. It is called the atlas of constant local trivializations.Two such atlases are said to be equivalent if their union also is an atlas ofconstant local trivializations. They are associated with the same horizontalfoliation.There is the 1–1 correspondence between the curvature–free connectionsΓ on a bundle Y −→ X and the equivalence classes of atlases Ψ c of constantlocal trivializations of Y such that Γ i α = 0 relative to the coordinates of thecorresponding atlas Ψ c [Canarutto (1986)].Connections on a bundle over a 1D base X 1 are curvature–free connections.In particular, let Y −→ X 1 be such a bundle (X 1 = R or X 1 = S 1 ). Itis coordinated by (t, y i ), where t is either the canonical parameter of R orthe standard local coordinate of S 1 together with the transition functionst ′ = t+const. Relative to this coordinate, the base X 1 admits the standardvector–field ∂ t and the standard one–form dt. Let Γ be a connection on Y−→ X 1 . Such a connection defines a horizontal foliation on Y −→ X 1 . Itsleaves are the integral curves of the horizontal liftτ Γ = ∂ t + Γ i ∂ i (5.37)of ∂ t by Γ. The corresponding Pfaffian system is locally generated by theforms (dy i − Γ i dt). There exists an atlas of constant local trivializations(t, y i ) such that Γ i = 0 and τ Γ = ∂ t relative to these coordinates.A connection Γ on Y → X 1 is called complete if the horizontal vector–field (5.37) is complete. Every trivialization of Y → R defines a completeconnection. Conversely, every complete connection on Y → R defines atrivialization Y ≃ R×M. The vector–field (5.37) becomes the vector–field ∂ t on R×M. As a proof, every trivialization of Y → R defines a 1–parameter group of isomorphisms of Y → R over Id R , and hence a completeconnection. Conversely, let Γ be a complete connection on Y → R. Thevector–field τ Γ (5.37) is the generator of a 1–parameter group G Γ which actsfreely on Y . The orbits of this action are the integral sections of τ Γ . Hencewe get a projection Y → M = Y/G Γ which, together with the projection