Generalized normal homogeneous Riemannian metrics on spheres and projective spaces
| Autor: | Berestovskii, V. N., Nikonorov, Yu. G. |
|---|---|
| Rok vydání: | 2012 |
| Předmět: | |
| Zdroj: | Ann.Glob.Anal.Geom., 45 (2014), 167-196 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s10455-013-9393-x |
| Popis: | In this paper we develop new methods of study of generalized normal homogeneous Riemannian manifolds. In particular, we obtain a complete classification of generalized normal homogeneous Riemannian metrics on spheres. We prove that for any connected (almost effective) transitive on $S^n$ compact Lie group $G$, the family of $G$-invariant Riemannian metrics on $S^n$ contains generalized normal homogeneous but not normal homogeneous metrics if and only if this family depends on more than one parameters. Any such family (that exists only for $n=2k+1$) contains a metric $g_{\can}$ of constant sectional curvature 1 on $S^n$. We also prove that $(S^{2k+1}, g_{\can})$ is Clifford-Wolf homogeneous, and therefore generalized normal homogeneous, with respect to $G$ (excepting the groups $G=SU(k+1)$ with odd $k+1$). The space of unit Killing vector fields on $(S^{2k+1}, g_{\can})$ from Lie algebra $\mathfrak{g}$ of Lie group $G$ is described as some symmetric space (excepting the case $G=U(k+1)$ when one obtains the union of all complex Grassmannians in $\mathbb{C}^{k+1}$). Comment: 32 pages |
| Databáze: | arXiv |
| Externí odkaz: |
|