Spectral Properties of Killing Vector Fields of Constant Length and Bounded Killing Vector Fields

Abstract

AbstractThis paper is a survey of recent results related to spectral properties of Killing vector fields of constant length and of some their natural generalizations on Riemannian manifolds. One of the main result is the following: If \(\mathfrak {g}\) is a Lie algebra of Killing vector fields on a given Riemannian manifold (M, g), and \(X\in \mathfrak {g}\) has constant length on (M, g), then the linear operator \( \operatorname {\mathrm {ad}}(X):\mathfrak {g} \rightarrow \mathfrak {g}\) has a pure imaginary spectrum (Nikonorov, J. Geom. Phys. 145 (2019), 103485). We discuss also more detailed structure results on the corresponding operator \( \operatorname {\mathrm {ad}}(X)\). Related results for geodesic orbit Riemannian spaces are considered. Finitely, we discuss some generalizations obtained recently by Xu and Nikonorov (Algebraic properties of bounded Killing vector fields. Asian J. Math. 2020 (accepted), see also. arXiv:1904.08710) for bounded Killing vector fields.KeywordsBounded Killing vector fieldsGeodesic orbit spaceHomogeneous Riemannian spaceKilling vector field of constant length2010 Mathematical Subject Classification53C2053C2553C30