Classifications of bundle connection pairs by parallel translation and lassos
Let E be a vector bundle over a smooth connected manifold M with typical fiber space V (V is a complex or real finite-dimensional vector space) and with structure group G (G is a closed Lie subgroup of \(G\ell (V))\). Pairs (E,V) and \((E',V')\) of connections on E are equivalent if there is a vector bundle isomorphism K: \(E\to E'\) with \(KV=V'K\) such that K respects the structure group G. The set of equivalence classes [E,V] will be denoted by \({\mathcal E}={\mathcal E}(M,V,G).\) Let \(\Omega_ m\) be the space of \(H^ 1\)-loops in M starting at \(m\in M\) and \({\mathcal M}^ a \)subset of \(C^{\infty}(\Omega_ m,G)\) where every \(P\in {\mathcal M}\subset C^{\infty}(\Omega_ m,G)\) is multiplicative \((P(\sigma \tau)=P(\sigma)\cdot P(\tau)\) for \(\sigma,\tau \in \Omega_ m\), \(\sigma\) \(\tau\) is the join of the loops \(\sigma\) and \(\tau)\), parametrization invariant \((P(\sigma ')=P(\sigma)\), if \(\sigma '\) is a reparametrization of \(\sigma)\) and strongly differentiable. Two functions \(P,P'\in {\mathcal M}\) are equivalent if for all \(\sigma \in \Omega_ m\), \(P'(\sigma)=g\cdot P(\sigma)\cdot g^{-1}\) for any \(g\in G\). The set of all equivalence classes [P] will be denoted by \(\bar {\mathcal M}=\bar {\mathcal M}(M,m,V,G).\) For a connection \(\nabla\) on E the parallel transfer operator on \(\Omega_ m\) is denoted by \(P^{\nabla}\) and the element of \(\bar {\mathcal M}\) defined by \([K\otimes P^{\nabla}(.)\otimes K^{-1}]\) by \([P^{\nabla}]\) where K: \(E_ m\to V\) is a linear isomorphism such that for every admissible local trivialization \(\psi\) of E, \(K\otimes \psi_ m^{-1}\in G\). One of the main results of this remarkable memoir is theorem 5.1 which states that the correspondence [E,\(\nabla]\in {\mathcal E}\to [P^{\nabla}]\in \bar {\mathcal M}\) is one to one. A similar statement was announced by \textit{S. Kobayashi} [C. R. Acad. Sci., Paris 238, 318-319 (1954; Zbl 0055.166)]. Another main result states that if M is simply connected then the set \({\mathcal E}\) is in one-to-one correspondence with the set of Lie algebra valued 1-forms on \(\Omega_ m\) (called integrated lassos). This result generalizes a result of \textit{A. Weil}: there is a bijection from U(1)- line bundle connection pairs to the set of curvature 2-forms [Variétés Kählériennes, Herman Paris 1958; Zbl 0137.411]. Also the notion of a lasso due to \textit{L. Gross} is generalized here [J. Funct. Anal. 63, 1-46 (1985; Zbl 0624.53021)].
scientific article; zbMATH DE number 4107606
scientific article