2019/04/18 by Ming Xu, Xu, Ming, Yu. G. Nikonorov +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.1904.08710
openalex publication_date 2019/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we consider a connected Riemannian manifold M where a connected Lie group G acts effectively and isometrically. Assume X∈\mathfrakg=Lie(G) defines a bounded Killing vector field, we find some crucial algebraic properties of the decomposition X=Xr+Xs according to a Levi decomposition \mathfrakg=\mathfrakr(\mathfrakg)+\mathfraks, where \mathfrakr(\mathfrakg) is the radical, and \mathfraks=\mathfraksc⊕\mathfraksnc is a Levi subalgebra. The decomposition X=Xr+Xs coincides with the abstract Jordan decomposition of X, and is unique in the sense that it does not depend on the choice of \mathfraks. By these properties, we prove that the eigenvalues of ad(X):\mathfrakg→\mathfrakg 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 \mathfrakg. We prove that the space of all these bounded Killing vector fields, or equivalently the space of all bounded vectors in \mathfrakg for G/H, is a compact Lie subalgebra, such that its semi-simple part is the ideal \mathfrakc_\mathfraksc(\mathfrakr(\mathfrakg)) of \mathfrakg, and its Abelian part is the sum of \mathfrakc_\mathfrakc(\mathfrakr(\mathfrakg)) (\mathfraksnc) and all two-dimensional irreducible ad(\mathfrakr(\mathfrakg))-representations in \mathfrakc_\mathfrakc(\mathfrakn)(\mathfraksnc) corresponding to nonzero imaginary weights, i.e. ℝ-linear functionals λ:\mathfrakr(\mathfrakg)→ \mathfrakr(\mathfrakg)/\mathfrakn(\mathfrakg) →ℝ√(-1), where \mathfrakn(\mathfrakg) is the nilradical.