Generalized normal homogeneous Riemannian metrics on spheres and projective spaces

TL;DR

This paper classifies generalized normal homogeneous Riemannian metrics on spheres, showing their relation to group actions, and identifies specific metrics with constant curvature and symmetry properties.

Contribution

It provides a complete classification of such metrics on spheres and characterizes their symmetry and parameter dependence, especially for odd-dimensional spheres.

Findings

Existence of non-normal homogeneous generalized normal homogeneous metrics depends on multiple parameters.

The standard metric on odd-dimensional spheres is Clifford-Wolf homogeneous and generalized normal homogeneous.

The space of unit Killing vector fields is described as a symmetric space, with exceptions noted.

Abstract

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 $\mathbb{C}^{k+1}$).