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On the Neifeld connection on a Grassmann manifold of Banach type. (Russian) Zbl 0930.53023
In 1976, E. G. Neifel’d proposed a way to construct a linear connection on a finite-dimensional Grassmann manifold \(G_{k+m,k}\) [Izv. Vyssh. Uchebn. Zaved., Mat. 1976, No. 11(174), 48-55 (1976; Zbl 0345.53003)] via a normalization, a mapping \(f : G_{k+m,m} \to G_{k+m,k}\). In the paper under review, the author considers an infinite-dimensional version of this construction.
For a Banach space \(E\), the author takes the Grassmann manifold \(G_{F,H}(E)\) of subspaces which are isomorphic to a subspace \(F \subset E\) and have a topological complement isomorphic to \(H \subset E\), and a normalization mapping \(f: G_{F,H} \to G_{H,F}\). Let \(\lambda(G_{F,H})\) be the universal vector bundle over \(G_{F,H}\) with fibre \(F\). Following Neifeld’s construction, the author demonstrates that the canonical flat connection on \(E\) induces via \(f\) linear connections \(\nabla'\) on \(\lambda(G_{F,H})\) and \(\nabla''\) on \(\lambda(G_{H,F})\). Then he proves that \(TG(F,H) \cong \lambda(G_{F,H}) \otimes f^*\lambda(G_{H,F})\), and thus obtains a linear connection \(\nabla\) on \(TG(F,H)\). Furthermore, he gives an expression for the operator of connection \(\nabla\) with respect to a canonical chart on \(G_{F,H}\).
MSC:
53C05 Connections, general theory
53C30 Differential geometry of homogeneous manifolds
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