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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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