Homogeneous Einstein metrics on Stiefel manifolds

Andreas Arvanitoyeorgos

Commentationes Mathematicae Universitatis Carolinae (1996)

  • Volume: 37, Issue: 3, page 627-634
  • ISSN: 0010-2628

Abstract

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A Stiefel manifold V k 𝐑 n is the set of orthonormal k -frames in 𝐑 n , and it is diffeomorphic to the homogeneous space S O ( n ) / S O ( n - k ) . We study S O ( n ) -invariant Einstein metrics on this space. We determine when the standard metric on S O ( n ) / S O ( n - k ) is Einstein, and we give an explicit solution to the Einstein equation for the space V 2 𝐑 n .

How to cite

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Arvanitoyeorgos, Andreas. "Homogeneous Einstein metrics on Stiefel manifolds." Commentationes Mathematicae Universitatis Carolinae 37.3 (1996): 627-634. <http://eudml.org/doc/247909>.

@article{Arvanitoyeorgos1996,
abstract = {A Stiefel manifold $V_k\mathbf \{R\}^n$ is the set of orthonormal $k$-frames in $\mathbf \{R\}^n$, and it is diffeomorphic to the homogeneous space $SO(n)/SO(n-k)$. We study $SO(n)$-invariant Einstein metrics on this space. We determine when the standard metric on $SO(n)/SO(n-k)$ is Einstein, and we give an explicit solution to the Einstein equation for the space $V_2\mathbf \{R\}^n$.},
author = {Arvanitoyeorgos, Andreas},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Riemannian geometry; homogeneous spaces; Einstein metrics; Stiefel manifolds; homogeneous space; homogeneous Riemannian metric; Einstein metric; Stiefel manifold},
language = {eng},
number = {3},
pages = {627-634},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Homogeneous Einstein metrics on Stiefel manifolds},
url = {http://eudml.org/doc/247909},
volume = {37},
year = {1996},
}

TY - JOUR
AU - Arvanitoyeorgos, Andreas
TI - Homogeneous Einstein metrics on Stiefel manifolds
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1996
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 37
IS - 3
SP - 627
EP - 634
AB - A Stiefel manifold $V_k\mathbf {R}^n$ is the set of orthonormal $k$-frames in $\mathbf {R}^n$, and it is diffeomorphic to the homogeneous space $SO(n)/SO(n-k)$. We study $SO(n)$-invariant Einstein metrics on this space. We determine when the standard metric on $SO(n)/SO(n-k)$ is Einstein, and we give an explicit solution to the Einstein equation for the space $V_2\mathbf {R}^n$.
LA - eng
KW - Riemannian geometry; homogeneous spaces; Einstein metrics; Stiefel manifolds; homogeneous space; homogeneous Riemannian metric; Einstein metric; Stiefel manifold
UR - http://eudml.org/doc/247909
ER -

References

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  2. Berger M., Quelques formules de variation pour une structure riemannienne, Ann. Sci. Éc. Norm. Sup. 3 (1970), 285-294. (1970) Zbl0204.54802MR0278238
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  4. James I.M., The Topology of Stiefel Manifolds, Cambridge University Press, Lecture Note Series 24, Great Britain, 1976. Zbl0337.55017MR0431239
  5. Jensen G.R., Einstein metrics on principal fibre bundles, J. Diff. Geom. 8 (1973), 599-614. (1973) Zbl0284.53038MR0353209
  6. Kobayashi S., Topology of positively pinched Kähler manifolds, Tôhoku Math. J. 15 (1963), 121-139. (1963) Zbl0114.37601MR0154235
  7. Kobayashi S., Nomizu K., Foundations of Differential Geometry, Vol II, Interscience, New York, 1969. Zbl0526.53001MR0238225
  8. Kowalski O., Vlášek Z., Homogeneous Einstein metrics on Aloff-Wallach spaces, Diff. Geom. Appl. 3 (1993), 157-167. (1993) MR1243541
  9. Sagle A.A., Some homogeneous Einstein manifolds, Nagoya Math. J. 39 (1970), 81-106. (1970) Zbl0198.54801MR0271867
  10. Wang M., Some examples of homogeneous Einstein manifolds in dimension seven, Duke Math. J. 49.1 (1982), 23-28. (1982) Zbl0488.53035MR0650366
  11. Wang M., Ziller W., On normal homogeneous Einstein metrics, Ann. Sci. Éc. Norm. Sup. 18 (1985), 563-633. (1985) MR0839687

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