Dynamical systems with an invariant space of vector fields

Abstract

1. In the theory of differentiable dynamical systems, the natural actions on coset spaces are of particular interest, not least because the general situation is so intractable. Let G be a Lie group and H a closed subgroup such that G/H is compact. A natural action on G/H is a group of transformations of G/H of the form xH * aa(x)H, where a is an element of G and a is an automorphism of G such that c(H)=H. The space of right-invariant vector fields on G is carried by the natural projection ir, x * xH, onto a space of vector fields on G/H, and this space is invariant under each natural action. Suppose now that H is discrete. Then G is unimodular. A Haar measure on G determines a finite Borel measure on G/H invariant under each action. Analogously, a translation invariant n-form w on G, where n is the dimension of G, determines an n-form -q on G/H, and T*y7 equals + -i for each natural transformation T. It is such a phenomenon that we examine in this paper: A dynamical system in which a certain finite-dimensional linear space of vector fields and a certain differential form are (essentially) invariant under the action. In the next few paragraphs we define several expressions, and then state our main result. The following section is devoted to a proof of this theorem. Additional results are presented in the third section. Let M be a differentiable manifold and let X be a finite-dimensional linear space of vector fields on M. We say that X is spanning if, for each p in M, the evaluation atp, X -? Xp, maps X onto the tangent space of M atp. If the dimension of X equals the dimension of M, in which case these mappings are each one-to-one, then X is said to be simply spanning. If X is simply spanning, an affine connection V on M is determined by the condition: