Asian Journal of Mathematics

Volume 25 (2021)

Number 2

Algebraic properties of bounded Killing vector fields

Pages: 229 – 242



Ming Xu (School of Mathematical Sciences, Capital Normal University, Beijing, China)

Yu. G. Nikonorov (Southern Mathematical Institute of the Vladikavkaz Scientific Centre of the Russian Academy of Sciences, Vladikavkaz, Russia)


In this paper, we consider a connected Riemannian manifold $M$ where a connected Lie group $G$ acts effectively and isometrically. Assume $X \in \mathfrak{g} = \operatorname{Lie}(G)$ defines a bounded Killing vector field, we find some crucial algebraic properties of the decomposition $X = X_r + X_s$ according to a Levi decomposition $\mathfrak{g} = \mathfrak{r} (\mathfrak{g}) + \mathfrak{s}$, where $\mathfrak{rg}$ is the radical, and $\mathfrak{s} = {\mathfrak{s}_c \oplus \mathfrak{s}_{nc}}$ is a Levi subalgebra. The decomposition $X = X_r + X_s$ coincides with the abstract Jordan decomposition of $X$, and is unique in the sense that it does not depend on the choice of $\mathfrak{s}$. By these properties, we prove that the eigenvalues of $\operatorname{ad} (X) : \mathfrak{g} \to \mathfrak{g}$ are all imaginary. Furthermore, when $M = G/H$ is a Riemannian homogeneous space, we can completely determine all bounded Killing vector fields induced by vectors in $\mathfrak{g}$. We prove that the space of all these bounded Killing vector fields, or equivalently the space of all bounded vectors in $\mathfrak{g}$ for $G/H$, is a compact Lie subalgebra, such that its semi-simple part is the ideal $\mathfrak{c}_{\mathfrak{s}_c} (\mathfrak{r}(\mathfrak{g}))$ of $\mathfrak{g}$, and its Abelian part is the sum of $\mathfrak{c}_{\mathfrak{c} (\mathfrak{r} (\mathfrak{g}))} (\mathfrak{s}_{nc})$ and all two-dimensional irreducible $\operatorname{ad} (\mathfrak{r}(\mathfrak{g}))$-representations in $\mathfrak{c}_{\mathfrak{c}(n)} (\mathfrak{s}_{nc})$ corresponding to nonzero imaginary weights, i.e. $\mathbb{R}$-linear functionals $\lambda : \mathfrak{r}(\mathfrak{g}) \to \mathfrak{r}(\mathfrak{g}) / n(\mathfrak{g}) \to \mathbb{R} \sqrt{-1}$, where $n(\mathfrak{g})$ is the nilradical.


bounded Killing vector field, Killing vector field of constant length, bounded vector for a coset space, Levi decomposition, Levi subalgebra, nilradical, radical

2010 Mathematics Subject Classification

22E46, 53C20, 53C30

The first-named author is supported by National Natural Science Foundation of China (No. 12131012, No. 11771331, No. 11821101), Beijing Natural Science Foundation (No. Z180004), and Capacity Building for Sci-Tech Innovation – Fundamental Scientific Research Funds (No. KM201910028021).

Received 29 April 2020

Accepted 28 July 2020

Published 15 October 2021