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- W2024447621 abstract "Let M be a connected manifold and G be a closed Lie subgroup of GL(V)—the general linear group on a finite dimensional vector space V. Denote by (Ωm) the space of H1-loops in M starting at a fixed point (m). Let M be the set of P ϵ C∞(Ωm, G), modulo conjugation by an element of G, such that P(στ) = P(σ) P(τ) for σ, τ ϵ Ωm (στ is the concatenation of the loops σ and τ), P(σ′) = P(σ) if σ′ is reparameterization of σ, and the differential of P satisfies a “locality” condition. It is shown that the bundle connection pairs (E, ▽) (up to equivalence), with structure group G and fiber V, are in one to one correspondence with M—a similar result has been announced by Kobayashi. The correspondence is induced by the parallel translation operators of connections. Furthermore, if the manifold (M) is simply connected, then the space of bundle connection pairs can be classified by a collection of Lie algebra valued 1-forms on the manifold Ωm (called integrated lassos). These 1-forms are related to the differentials of elements of M. This last result generalizes Weil's characterization of U(1)—line bundle connection pairs by the curvature 2-form. It is also a generalization of Gross' results to base manifolds (M) other than Rn." @default.
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- W2024447621 date "1989-03-01" @default.
- W2024447621 modified "2023-09-29" @default.
- W2024447621 title "Classifications of bundle connection pairs by parallel translation and lassos" @default.
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- W2024447621 doi "https://doi.org/10.1016/0022-1236(89)90035-9" @default.
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