不变刚性结构和不变几乎刚性结构的不存在性
Nonexistence of invariant rigid structures and invariant almost rigid structures
摘要：
证明了李群及其格的某些保体积作用不保持Gromov意义下的刚性几何结构。所考虑的行动是“异国情调”的例子，由Katok和刘易斯和第一作者，通过炸毁闭轨道在齐性空间上的著名行动。齐性空间上的作用都保持仿射联络，而沿例外因子的作用保持投射结构。这些结构不能以某种方式“粘合在一起”以在整个空间上提供刚性结构的事实并不明显。我们还定义了几乎刚性结构的概念。刚性结构的典型例子是一个整体框架，而几乎刚性结构的典型例子是一个沿着某个例外因子退化的框架。我们表明，上述讨论的行动具有不变的几乎刚性的结构。Gromov证明了在拓扑传递群作用下具有刚性几何结构不变的流形在开稠密集上是齐次的。这个开稠密集如何一般地被当作整个流形是一个重要的问题，有许多动力学应用。我们的结果指出了一种方法，在这种方法中，几何结构不能退化开稠密集。
英文摘要：
We prove that certain volume preserving actions of Lie groups and their lattices do not preserve rigid geometric structures in the sense of Gromov. The actions considered are the "exotic" examples obtained by Katok and Lewis and the first author, by blowing up closed orbits in the well known actions on homogeneous spaces. The actions on homogeneous spaces all preserve affine connections, whereas the action along the exceptional divisor preserves a projective structure. The fact that these structures cannot in some way be "glued together" to give a rigid structure on the entire space is not obvious. We also define the notion of an almost rigid structure. The paradigmatic example of a rigid structure is a global framing and the paradigmatic example of an almost rigid structure is a framing that is degenerate along some exceptional divisor. We show that the actions discussed above do possess an invariant almost rigid structure. Gromov has shown that a manifold with rigid geometric structure invariant under a topologically transitive group action is homogeneous on an open dense set. How generally this open dense set can be taken to be the entire manifold is an important question with many dynamical applications. Our results indicate one way in which the geometric structure cannot degenerate off the open dense set.
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