标题： 向量场的连接稳定性及其在爱因斯坦-福克方程组天体物理应用 显示英文标题

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

摘要： 在本文中，我们提出了一种方法，用于考虑光滑流形上可能不紧致的光滑向量场的稳定性。我们表明，这种称为“连接稳定性”的稳定性在紧致流形的情况下等价于结构稳定性。我们证明，如果X是一个连接稳定的向量场，那么它的任何非零标量乘积也是一个连接稳定的向量场。我们给出了一个非紧致流形上的连接稳定向量场的例子，并且我们还表明谐振子不是一个连接稳定的向量场。我们提出了一种技术来证明一类向量场不是连接稳定的。作为具体的物理例子，我们将分析应用于天体物理背景下的爱因斯坦-维拉索系统，我们提出了一种方法，原则上部分地有助于理解星系旋转曲线问题。 显示英文摘要

摘要： 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.