Transitive riemannian isometry groups with nilpotent radicals

Gordon, C.

Annales de l'Institut Fourier, Tome 31 (1981), p. 193-204 / Harvested from Numdam

Résumé

Étant donné un groupe de Lie connexe $G$, dont le radical est nilpotent et qui opère transitivement par isométries sur un espace homogène riemannien $M$, on décrit la structure du plus grand groupe connexe $A$ des isométries de $M$ et l’inclusion de $G$ dans $A$. En conséquence, on obtient une condition suffisante pour que $G$ soit normal dans $A$. Dans le cas spécial d’une action simplement transitive de $G$ sur $M$, on construit un sous-groupe $G^{\prime }$ normal dans $A$, transitif sur $M$ et ayant la même dimension que $G$, et on donne une condition suffisante pour que $G^{\prime }$ soit localement isomorphe à $G$.

Abstract

Given that a connected Lie group $G$ with nilpotent radical acts transitively by isometries on a connected Riemannian manifold $M$, the structure of the full connected isometry group $A$ of $M$ and the imbedding of $G$ in $A$ are described. In particular, if $G$ equals its derived subgroup and its Levi factors are of noncompact type, then $G$ is normal in $A$. In the special case of a simply transitive action of $G$ on $M$, a transitive normal subgroup $G^{\prime }$ of $A$ is constructed with $\dim G^{\prime } = \dim G$ and a sufficient condition is given for local isomorphism of $G^{\prime }$ and $G$.

Détail

Publié le : 1981-01-01

MR 82i:53040 | Zbl 0441.53034

DOI : https://doi.org/10.5802/aif.835

Bibliographie

[1] R. Azencott and E. N. Wilson, Homogeneous manifolds with negative curvature, Part I, Trans. Amer. Math. Soc., 215 (1976), 323-362. | MR 52 #15308 | Zbl 0293.53017

[2] R. Azencott and E. N. Wilson, Homogeneous manifolds with negative curvature, Part II, Mem. Amer. Math. Soc., 8 (1976). | MR 54 #13951 | Zbl 0355.53026

[3] C. Gordon, Riemannian isometry groups containing transitive reductive subgroups, Math. Ann., 248 (1980), 185-192. | MR 81e:53030 | Zbl 0412.53026

[4] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Academic Press, New York, 1978. | Zbl 0451.53038

[5] N. Jacobson, Lie algebras, Wiley Interscience, New York, 1962. | Zbl 0121.27504

[6] A. L. Oniščik, Inclusion relations among transitive compact transformation groups, Amer. Math. Soc. Transl., 50 (1966), 5-58. | Zbl 0207.33604

[7] H. Ozeki, On a transitive transformation group of a compact group manifold, Osaka J. Math., 14 (1977), 519-531. | MR 57 #1362 | Zbl 0382.57020

[8] E. N. Wilson, Isometry groups on homogeneous nilmanifolds, to appear in Geometriae Dedicata. | Zbl 0489.53045