Hiroshima Mathematical Journal

Geodesic orbit manifolds and Killing fields of constant length

Yuriĭ G. Nikonorov

Full-text: Open access

Abstract

The goal of this paper is to clarify connections between Killing fields of constant length on a Rimannian geodesic orbit manifold $(M,g)$ and the structure of its full isometry group. The Lie algebra of the full isometry group of $(M,g)$ is identified with the Lie algebra of Killing fields $\mathfrak{g}$ on $(M,g)$. We prove the following result: If $\mathfrak{a}$ is an abelian ideal of $\mathfrak{g}$, then every Killing field $X\in \mathfrak{a}$ has constant length. On the ground of this assertion we give a new proof of one result of C. Gordon: Every Riemannian geodesic orbit manifold of nonpositive Ricci curvature is a symmetric space.

Article information

Source
Hiroshima Math. J. Volume 43, Number 1 (2013), 129-137.

Dates
First available in Project Euclid: 10 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1368217953

Mathematical Reviews number (MathSciNet)
MR3066528

Zentralblatt MATH identifier
1276.53046

Subjects
Primary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]
Secondary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C35: Symmetric spaces [See also 32M15, 57T15]

Keywords
Killing fields of constant length homogeneous Riemannian manifolds geodesic orbit spaces symmetric spaces Ricci curvature

Citation

Nikonorov, Yuriĭ G. Geodesic orbit manifolds and Killing fields of constant length. Hiroshima Math. J. 43 (2013), no. 1, 129--137.https://projecteuclid.org/euclid.hmj/1368217953


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References

  • D.V. Alekseevsky, A. Arvanitoyeorgos, Riemannian flag manifolds with homogeneous geodesics, Trans. Amer. Math. Soc., 359 (2007), 3769–3789.
  • D.V. Alekseevsky, Yu.G. Nikonorov, Compact Riemannian manifolds with homogeneous geodesics, SIGMA Symmetry Integrability Geom. Methods Appl., 5 (2009), 093, 16 pages.
  • D.N. Akhiezer, È.B. Vinberg, Weakly symmetric spaces and spherical varieties, Transform. Groups, 4 (1999), 3–24.
  • V.N. Berestovskii, Yu.G. Nikonorov, On $\delta$-homogeneous Riemannian manifolds, Differential Geom. Appl., 26(5) (2008), 514–535.
  • V.N. Berestovskii, Yu.G. Nikonorov, On $\delta$-homogeneous Riemannian manifolds, II, Sib. Math. J., 50(2) (2009), 214–222.
  • V.N. Berestovskii, Yu.G. Nikonorov, Killing vector fields of constant length on Riemannian manifolds, Sib. Math. J., 49 (3) (2008) 395–407.
  • V.N. Berestovskii, Yu.G. Nikonorov, Clifford-Wolf homogeneous Riemannian manifolds, J. Differential Geom., 82(3), 2009, 467–500.
  • A.L. Besse, Einstein Manifolds, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1987.
  • Z. Du\u sek, O. Kowalski, S. Nikčević, New examples of Riemannian g. o. manifolds in dimension 7. Differential Geom. Appl., 21 (2004), 65–78.
  • C. Gordon, Homogeneous Riemannian manifolds whose geodesics are orbits, 155–174. In: Progress in Nonlinear Differential Equations. V. 20. Topics in geometry: in memory of Joseph D'Atri. Birkhäuser, 1996.
  • M. Goto, F.D. Grosshans, Semisimple Lie algebras. Lecture Notes in Pure and Applied Mathematics, Vol. 38. Marcel Dekker, Inc., New York-Basel, 1978.
  • S. Helgason, Differential geometry and symmetric spaces, Academic Press Inc., New-York, 1962.
  • S. Kobayashi, K. Nomizu, Foundations of differential geometry, Vol. I – A Wiley-Interscience Publication, New York, 1963; Vol. II – A Wiley-Interscience Publication, New York, 1969.
  • O. Kowalski, L. Vanhecke, Riemannian manifolds with homogeneous geodesics, Boll. Un. Mat. Ital. B (7), 5(1) (1991), 189–246.
  • H. Tamaru, Riemannian g. o. spaces fibered over irreducible symmetric spaces, Osaka J. Math., 36(4) (1999), 835–851.
  • H. Tamaru, Riemannian geodesic orbit metrics on fiber bundles, Algebras Groups Geom., 15(1) (1998), 55–67.
  • J.A. Wolf, Spaces of constant curvature, Publish or Perish, Inc., Wilmington, Delaware (U.S.A.), 1984.
  • J.A. Wolf, Harmonic Analysis on Commutative Spaces, American Mathematical Society, 2007.
  • K. Yano, S. Bochner, Curvature and Betti numbers, Princeton, New Jersey, Princeton University Press, 1953.