Compact Riemannian Manifolds with Homogeneous Geodesics

Dmitrii V. Alekseevskii ; Yurii G. Nikonorov

Symmetry, integrability and geometry: methods and applications, Tome 5 (2009)

Résumé

A homogeneous Riemannian space $(M=G/H,g)$ is called a geodesic orbit space (shortly, GO-space) if any geodesic is an orbit of one-parameter subgroup of the isometry group $G$. We study the structure of compact GO-spaces and give some sufficient conditions for existence and non-existence of an invariant metric $g$ with homogeneous geodesics on a homogeneous space of a compact Lie group $G$. We give a classification of compact simply connected GO-spaces $(M=G/H,g)$ of positive Euler characteristic. If the group $G$ is simple and the metric $g$ does not come from a bi-invariant metric of $G$, then $M$ is one of the flag manifolds $M_1=SO(2n+1)/U(n)$ or $M_2=Sp(n)/U(1)\cdot Sp(n-1)$ and $g$ is any invariant metric on $M$ which depends on two real parameters. In both cases, there exists unique (up to a scaling) symmetric metric $g_0$ such that $(M,g_0)$ is the symmetric space $M=SO(2n+2)/U(n+1)$ or, respectively, $\mathbb{C}P^{2n-1}$. The manifolds $M_1$, $M_2$ are weakly symmetric spaces.

Détail

Keywords: homogeneous spaces, weakly symmetric spaces, homogeneous spaces of positive Euler characteristic, geodesic orbit spaces, normal homogeneous Riemannian manifolds, geodesics.

Bibliographie

[1] Alekseevsky D. V., Arvanitoyeorgos A., “Riemannian flag manifolds with homogeneous geodesics”, Trans. Amer. Math. Soc., 359 (2007), 3769–3789 | DOI | MR | Zbl

[2] Akhiezer D. N., Vinberg E. B., “Weakly symmetric spaces and spherical varieties”, Transform. Groups, 4 (1999), 3–24 | DOI | MR | Zbl

[3] Berestovskii V. N., Nikonorov Yu. G., “On $\delta$-homogeneous Riemannian manifolds”, Differential Geom. Appl., 26 (2008), 514–535 ; math.DG/0611557 | DOI | MR

[4] Berestovskii V. N., Nikonorov Yu. G., “On $\delta$-homogeneous Riemannian manifolds. II”, Siber. Math. J., 50:2 (2009), 214–222 | DOI | MR

[5] Berestovskii V. N., Nikitenko E. V., Nikonorov Yu. G., The classification of $\delta$-homogeneous Riemannian manifolds with positive Euler characteristic, arXiv:0903:0457

[6] Berger M., “Les variétés riemanniennes homogènes normales à courbure strictement positive”, Ann. Scuola Norm. Sup. Pisa (3), 15 (1961), 179–246 | MR | Zbl

[7] Berndt J., Kowalski O., Vanhecke L., “Geodesics in weakly symmetric spaces”, Ann. Global Anal. Geom., 15 (1997), 153–156 | DOI | MR | Zbl

[8] Besse A. L., Einstein manifolds, Springer-Verlag, Berlin, 1987 | MR

[9] Borel A., de Siebenthal J., “Les sous-groups fermés de rang maximum des groups de Lie clos”, Comment. Math. Helv., 23 (1949), 200–221 | DOI | MR | Zbl

[10] Cartan É., “Sur une classe remarquable d'espaces de Riemann”, Bull. Soc. Math. France, 54 (1926), 214–264 ; Cartan É., “Sur une classe remarquable d'espaces de Riemann. II”, Bull. Soc. Math. France, 55 (1927), 114–134 | MR | MR

[11] D'Atri J. E., Ziller W., “Naturally reductive metrics and Einstein metrics on compact Lie groups”, Mem. Amer. Math. Soc., 18 no 215, 1979, 1–72 | MR

[12] Dus̆ek Z., Kowalski O., Nikc̆ević S., “New examples of Riemannian g.o. manifolds in dimension 7”, Differential Geom. Appl., 21 (2004), 65–78 | DOI | MR | Zbl

[13] Gorbatzevich V. V., Onishchik A. L., Vinberg E. B., Lie groups and Lie algebras. III. Structure of Lie groups and Lie algebras, Encyclopaedia of Mathematical Sciences, 41, Springer-Verlag, Berlin, 1994 | MR

[14] Gordon C. S., “Homogeneous Riemannian manifolds whose geodesics are orbits”, Topics in Geometry: in Memory of Joseph D'Atri, Progr. Nonlinear Differential Equations Appl., 20, Birkhäuser Boston, Boston, MA, 1996, 155–174 | MR | Zbl

[15] Helgason S., Differential geometry and symmetric spaces, Pure and Applied Mathematics, 12, Academic Press, New York – London, 1962 | MR | Zbl

[16] Kaplan A., “On the geometry of groups of Heisenberg type”, Bull. London Math. Soc., 15 (1990), 32–42 | MR

[17] Kerr M. M., “Some new homogeneous Einstein metrics on symmetric spaces”, Trans. Amer. Math. Soc., 348 (1996), 153–171 | DOI | MR | Zbl

[18] Kobayashi S., Nomizu K., Foundations of differential geometry, Vol. I, John Wiley Sons, New York, 1963 ; Vol. II, 1969 | MR | Zbl

[19] Kostant B., “On holonomy and homogeneous spaces”, Nagoya Math. J., 12 (1957), 31–54 | MR | Zbl

[20] Kowalski O., Vanhecke L., “Riemannian manifolds with homogeneous geodesics”, Boll. Un. Mat. Ital. B (7), 5 (1991), 189–246 | MR | Zbl

[21] Onishchik A. L., Topology of transitive transformation groups, Johann Ambrosius Barth Verlag GmbH, Leipzig, 1994 | MR | Zbl

[22] Selberg A., “Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces, with applications to Dirichlet series”, J. Indian Math. Soc. (N.S.), 20 (1956), 47–87 | MR | Zbl

[23] Tamaru H., “Riemannin G. O. spaces fibered over irreducible symmetric spaces”, Osaka J. Math., 15 (1998), 835–851 | MR

[24] Tamaru H., “Riemannin geodesic orbit space metrics on fiber bundles”, Algebra Groups Geom., 36 (1999), 835–851 | MR | Zbl

[25] Wolf J. A., Spaces of constant curvature, Publish or Perish, Inc., Houston, TX, 1984 | MR

[26] Wolf J. A., Harmonic analysis on commutative spaces, Mathematical Surveys and Monographs, 142, American Mathematical Society, Providence, RI, 2007 | MR | Zbl

[27] Ziller W., “Weakly symmetric spaces”, Topics in Geometry: in Memory of Joseph D'Atri, Progr. Nonlinear Differential Equations Appl., 20, Birkhäuser Boston, Boston, MA, 1996, 355–368 | MR | Zbl

[28] Yakimova O. S., “Weakly symmetric Riemannian manifolds with a reductive isometry group”, Sb. Math., 195 (2004), 599–614 | DOI | MR | Zbl