## Hiroshima Mathematical Journal

### Geodesic orbit manifolds and Killing fields of constant length

Yuriĭ G. Nikonorov

#### 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

https://projecteuclid.org/euclid.hmj/1368217953

Mathematical Reviews number (MathSciNet)
MR3066528

Zentralblatt MATH identifier
1276.53046

#### 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

#### 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.