Annals of Global Analysis and Geometry

, Volume 45, Issue 3, pp 167–196 | Cite as

Generalized normal homogeneous Riemannian metrics on spheres and projective spaces

  • Valeriĭ Nikolaevich Berestovskiĭ
  • Yuriĭ Gennadievich Nikonorov


In this paper, we develop new methods to study generalized normal homogeneous Riemannian manifolds. In particular, we obtain a complete classification of generalized normal homogeneous Riemannian metrics on spheres \({S^n}\). We prove that for any connected (almost effective) transitive on \(S^n\) compact Lie group \(G\), the family of \(G\)-invariant Riemannian metrics on \(S^n\) contains generalized normal homogeneous but not normal homogeneous metrics if and only if this family depends on more than one parameters and \(n\ge 5\). Any such family (that exists only for \(n=2k+1\)) contains a metric \(g_\mathrm{can}\) of constant sectional curvature \(1\) on \(S^n\). We also prove that \((S^{2k+1}, g_\mathrm{can})\) is Clifford–Wolf homogeneous, and therefore generalized normal homogeneous, with respect to \(G\) (except the groups \(G={ SU}(k+1)\) with odd \(k+1\)). The space of unit Killing vector fields on \((S^{2k+1}, g_\mathrm{can})\) from Lie algebra \(\mathfrak g \) of Lie group \(G\) is described as some symmetric space (except the case \(G=U(k+1)\) when one obtains the union of all complex Grassmannians in \(\mathbb{C }^{k+1}\)).


Clifford algebras Clifford–Wolf homogeneous spaces  Generalized normal homogeneous Riemannian manifolds g.o. spaces Grassmannian algebra Homogeneous spaces Hopf fibrations Normal homogeneous Riemannian manifolds Generalized normal homogeneous but not normal homogeneous Riemannian submersions (weakly) symmetric spaces 

Mathematics Subject Classification (2000)

Primary 53C20 Secondary 53C25 53C35 



We thank Professor Wolfgang Ziller for useful discussions and Natalia Berestovskaya for help in preparation of the text. The authors are indebted to the anonymous referee for helpful comments and suggestions that improved the presentation of this paper.


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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Valeriĭ Nikolaevich Berestovskiĭ
    • 1
  • Yuriĭ Gennadievich Nikonorov
    • 2
  1. 1.Omsk Branch of Sobolev Institute of Mathematics of SD RASOmskRussia
  2. 2.South Mathematical Institute of VSC RASVladikavkazRussia

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