Algebraic properties of bounded Killing vector fields
| Autor: | Yu. G. Nikonorov, Ming Xu |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Mathematics - Differential Geometry
Applied Mathematics General Mathematics Subalgebra Lie group Riemannian manifold Levi decomposition Combinatorics Killing vector field Differential Geometry (math.DG) Homogeneous space FOS: Mathematics Ideal (ring theory) Abelian group Mathematics::Representation Theory Mathematics |
| Zdroj: | Asian Journal of Mathematics. 25:229-242 |
| ISSN: | 1945-0036 1093-6106 |
| Popis: | 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 $\mathrm{ad}(X):\mathfrak{g}\rightarrow\mathfrak{g}$ 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 $\lambda:\mathfrak{r}(\mathfrak{g})\rightarrow \mathfrak{r}(\mathfrak{g})/\mathfrak{n}(\mathfrak{g}) \rightarrow\mathbb{R}\sqrt{-1}$, where $\mathfrak{n}(\mathfrak{g})$ is the nilradical. |
| Databáze: | OpenAIRE |
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