Transitive riemannian isometry groups with nilpotent radicals

Gordon, C.

Annales de l'Institut Fourier, Tome 31 (1981) no. 2, pp. 193-204.

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 $G$.

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 $dim{G}^{\prime }=dimG$ and a sufficient condition is given for local isomorphism of $G^{\prime }$ and $G$.

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