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Infinite-dimensional homogeneous manifolds. (English) Zbl 0814.58007

First, the author generalizes the Cartan-Ambrose-Hicks theorem to smooth Banach manifolds. Using it he characterizes the simply-connected symmetric manifolds as smooth Banach manifolds admitting a geodesically complete torsion-free affine connection whose curvature tensor is parallel. Then he proves an integrability theorem for invariant almost complex structures on any symmetric Banach manifold that admits an invariant Finsler structure. Various consequences of this are discussed.

MSC:

58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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