Algebraic properties of bounded Killing vector fields

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.

Keywords

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

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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