The Induced Connections on Total Spaces of Fibred Manifolds

Let Y → M be a fibred manifold with m-dimensional base and n-dimensional fibres. If m > 2 and n > 3, we classify all linear connections A(Γ,Λ,Θ) : T Y → J(T Y → Y ) in T Y → Y (i.e., classical linear connections on Y ) depending canonically on a system (Γ,Λ,Θ) consisting of a general connection Γ : Y → JY in Y → M , a torsion free classical linear connection Λ : T M → J(T M → M) on M and a linear connection Θ : V Y → J(V Y → Y ) in the vertical bundle V Y → Y . Introduction All manifolds considered in the paper are assumed to be Hausdorff, second countable, without boundary, finite dimensional and smooth (of class C). Maps between manifolds are assumed to be smooth (infinitely differentiable). Let Y → M be a fibred manifold withm-dimensional baseM and n-dimensional fibres. Let Γ : Y → J1Y be a general connection in a fibred manifold Y → M (i.e., a section of the first jet prolongation π1 0 : J 1Y → Y of Y → M), Λ : TM → J1(TM → M) be a torsion free linear connection in the tangent bundle TM → M of M (i.e., a torsion free classical linear connection on M) and Θ : V Y → J1(V Y → Y ) be a linear connection in the vertical bundle V Y → Y of Y → M (i.e., a vertical classical linear connection on Y → M). More on connections can be found in [6]. Here we study how to construct canonically a linear connection A(Γ,Λ,Θ) : TY → J1(TY → Y ) in TY → Y (i.e., a classical linear connection on the total space Y ) from the system (Γ,Λ,Θ) as above. For example, one can construct a linear connection Ψ = Ψ(Γ,Λ,Θ) : TY → J1(TY → Y ) in TY → Y as follows. We decompose Z ∈ TyY into the horizontal part h(Z) = Γ(y, Z0), Z0 ∈ TxM , x = p(y) and the vertical part vZ. We take a vector field X on M such that j1 xX = Λ(Z0) and construct its Γ-lift ΓX : Y → TY , 2010 Mathematics Subject Classification: Primary 53C05; Secondary 58A32.