Algebraic properties of bounded Killing vector fields

Abstract

In this paper, we consider a connected Riemannian manifold $M$ where a connected Lie group $G$ acts effectively and isometrically. Assume $X\in \mathfrak{g}=\mathrm{Lie}(G)$ 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 $\mathfrak{r}(\mathfrak{g})$ 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 $\mathrm{ad}(X):\mathfrak{g}\to \mathfrak{g}$ 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 $\mathrm{ad}(\mathfrak{r}(\mathfrak{g}))$-representations in ${\mathfrak{c}}_{\mathfrak{c}(\mathfrak{n})}({\mathfrak{s}}_{nc})$ corresponding to nonzero imaginary weights, i.e. $\mathbb{R}$-linear functionals $\lambda :\mathfrak{r}(\mathfrak{g})\to \mathfrak{r}(\mathfrak{g})/\mathfrak{n}(\mathfrak{g})\to \mathbb{R}\sqrt{-1}$, where $\mathfrak{n}(\mathfrak{g})$ is the nilradical.

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