Prolongation of connections to bundles of infinitely near points

Morimoto, Akihiko

doi:10.4310/jdg/1214433720

Cited by 55 publications

(44 citation statements)

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“…A lifting (the so-called complete lifting) of connections to Weil bundles was constructed long ago in [5]. But the problem of finding all such liftings is still unsolved and seems to be hard.…”

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confidence: 99%

“…This paper contains a classification of all affine liftings of torsion-free linear connections on n-dimensional manifolds to any linear connections on Weil bundles under the condition that n ≥ 3.A lifting (the so-called complete lifting) of connections to Weil bundles was constructed long ago in [5]. But the problem of finding all such liftings is still unsolved and seems to be hard.…”

mentioning

confidence: 99%

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Affine liftings of torsion-free connections to Weil bundles

Dębecki

2009

Colloq. Math.

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Abstract. This paper contains a classification of all affine liftings of torsion-free linear connections on n-dimensional manifolds to any linear connections on Weil bundles under the condition that n ≥ 3.A lifting (the so-called complete lifting) of connections to Weil bundles was constructed long ago in [5]. But the problem of finding all such liftings is still unsolved and seems to be hard. The recent result [2] on linear liftings of symmetric tensor fields of type (1, 2) to Weil bundles enables us to cope with a very special case of this problem, namely finding all affine liftings of torsion-free connections to Weil bundles.Let A be a Weil algebra inducing the Weil functor T A (see [4]) and let n be a non-negative integer. We will denote by Co M the set of all linear connections on a manifold M and by ToFrCo M the set of all torsion-free linear connections on M.A lifting of torsion-free linear connections to linear connections on T A is, by definition, a family of maps L M : ToFrCo M → Co T A M indexed by all n-dimensional manifolds and satisfying (1)

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confidence: 99%

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confidence: 99%

Affine liftings of torsion-free connections to Weil bundles

Dębecki

2009

Colloq. Math.

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“…Some of them were applied in constructions on connections (see e.g. [3], [7]). In Section 8, using the flow operator W r we construct a principal connection W r (Γ, Λ) on the principal bundle W r P → P .…”

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confidence: 99%

“…To obtain such constructions, some vector fields obtained from right-invariant vector fields are useful (see e.g. [7] in the case of the trivial Lie group G = {e}). Taking into account the above remarks, we see that a complete classification of all PB m (G)-gauge-natural operators A : T G-inv T W r in the sense of Definition 1 is especially important.…”

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confidence: 99%

Lifting right-invariant vector fields and prolongation of connections

Mikulski

2009

Ann. Polon. Math.

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Abstract. We describe all PBm(G)-gauge-natural operators A lifting right-invariant vector fields X on principal G-bundles P → M with m-dimensional bases into vector fields A(X) on the rth order principal prolongation W r P = P r M ×M J r P of P → M . In other words, we classify all PBm(G)-natural transformations J r LP ×M W r P → T W r P = LW r P ×M W r P covering the identity of W r P , where J r LP is the r-jet prolongation of the Lie algebroid LP = T P/G of P , i.e. we find all PBm(G)-natural transformations which are similar to the Kumpera-Spencer isomorphism J r LP = LW r P . We formulate axioms which characterize the flow operator of the gauge-bundle W r P → M . We apply the flow operator to prolongations of connections.0. Introduction. Let G be a Lie group with Lie algebra L(G) and e ∈ G be the unit element. Let PB m (G) denote the category of all principal G-bundles with m-dimensional bases and their (local) principal bundle isomorphisms over the identity group homomorphism.Let Φ, Ψ : P → Q be PB m (G)-maps. Let x ∈ M . The following conditions are equivalent: (i) j r p 0 Φ = j r p 0 Ψ for some p 0 ∈ P x ; (ii) j r p Φ = j r p Ψ for any p ∈ P x . We write j r x Φ = j r x Ψ if these conditions are satisfied (see [3]). Let P → M be a principal G-bundle with m-dimensional basis. Its rth principal prolongation W r P is defined as the space of all r-jets j r 0 ϕ of local trivializations ϕ : R m × G → P . By [3], W r P → M is a principal bundle with the structure group W r m G := J r 0 (R m × G, R m × G) 0 , and the fibred manifold W r P → M coincides with the fibred product P r M × M J r P , whereEvery PB m (G)-map Φ : P → Q can be extended (via composition of jets) to a principal bundle (local) isomorphism W r Φ :2000 Mathematics Subject Classification: 58A20, 58A32.

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“…for every n-dimensional manifold M and every linear connection ∇ on M , where ∇ A denotes the complete lift of ∇ (see [14], [7]), T and R the torsion and curvature tensors of ∇ and G ′ H ′′…”

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Linear liftings of skew symmetric tensor fields of type (1, 2) to Weil bundles

Dębecki

2010

Czech Math J

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Prolongation of connections to bundles of infinitely near points

Cited by 55 publications

References 5 publications

Affine liftings of torsion-free connections to Weil bundles

Affine liftings of torsion-free connections to Weil bundles

Lifting right-invariant vector fields and prolongation of connections

Linear liftings of skew symmetric tensor fields of type (1, 2) to Weil bundles

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