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A note on smooth toral reductions of spheres. (English) Zbl 0913.53020

The main result of this paper is the following Theorem. Let \(S\) be a 3-Sasakian manifold obtained by 3-Sasakian reduction of \(S^{4n-1}\) by a torus \(T^k\), and assume that \(S\) is neither a sphere nor a smooth quotient of a sphere by a finite group. If \(k>1\), then \(\dim S=7, 11, 15\). In addition, if \(k>4\), then \(\dim S=7\).
This result extends previous results of the authors and E. G. Rees [Invent. Math. 131, 321-344 (1998; Zbl 0901.53033); Balkan J. Geom. Appl. 1, 1-7 (1996; Zbl 0889.53030)].

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
57S25 Groups acting on specific manifolds
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