Abstract
We discuss the possibility for a given homogeneous space M=G/H to be diffeomorphic to each term of a sequence (Gn/Hn) of coset spaces, where the Gn 's are connected real Lie groups of arbitrarily large (finite) dimensions. We prove that this possibility does indeed arise when M is the total space of a convenient circle bundle. Typical examples are provided by odd dimensional spheres of dimension at least 3. Our main tool for this result is the theory of connections on principal bundles.
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de la Harpe, P., Takens, F. Examples of manifolds which are homogeneous spaces of lie groups of arbitrarily large dimension. Manuscripta Math 15, 275–287 (1975). https://doi.org/10.1007/BF01168679
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DOI: https://doi.org/10.1007/BF01168679