Connection stability of vector fields with astrophysical application to the Einstein-Vlasov system

In this paper we present a method for considering the stability of smooth vector fields on a smooth manifold which may not be compact. We show that these kind of stability which is called "connection stability" is equivalent to the structural stability in the case of compact manifolds. We prove if X is a connection stable vector field, then any multiplication of it by a nonzero scalar is also a connection stable vector field. We present an example of a connection stable vector field on a noncompact manifold, and we also show that harmonic oscillator is not a connection stable vector field. We present a technique to prove a class of vector fields are not connection stable. As a concrete physical example, we will apply the analysis to the Einstein- Vlasov system in an astrophysical context in which we propose an approach that could, in principle and partially, help to understand the problem of galaxy rotation curves.