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 is an element of g = 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 g = r(g) +s, where r(g) is the radical, and s = s(c) circle plus 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 s. By these properties, we prove that the eigenvalues of ad(X) : g -> 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 g. We prove that the space of all these bounded Killing vector fields, or equivalently the space of all bounded vectors in g for G/H, is a compact Lie subalgebra, such that its semi-simple part is the ideal c(sc) (r(g)) of g, and its Abelian part is the sum of c(c)((r)((g)())) (s(nc)) and all two-dimensional irreducible ad(r(g))-representations in c(c(n)) (s(nc)) corresponding to nonzero imaginary weights, i.e. R-linear functionals lambda : r(g) -> r(g)/n(g) -> R root-1, where n(g) is the nilradical.