Foliated Rigidity and Dynamics of Space-Time

scott adams对洛伦兹流形的研究可能是出于物理考虑，因为爱因斯坦场方程决定了时空上的洛伦兹度量。根据F.~Klein的Erlanger程序，研究几何物体的等距群是一种方法，而人们早就知道阿仑兹流形的等距群是李群。此外，任何李群作用于自身（通过左平移）保持任何左不变洛伦兹度规。然而，正如最初由R.~Zimmer， M.~Gromov和n。~Kowalsky，如果人们对动力学要求哪怕是一点点的复杂性，那么承认洛伦兹作用的李群的列表就会变得相当有限。在过去的几年里，我的工作一直集中在量化哪些连接李群通过连接洛伦兹流形的等长来承认复杂的作用。由于“复杂”的定义不同，答案也不同，我打算在未来几年研究更多的这些变化。特别地，我想确定在连通的洛伦兹流形上存在局部忠实的、非驯服的、等距作用的连通的、非紧致的、简单李群的集合。我相信它应该是真的，任何这样的群要么有无限的中心，要么局部同构于李群的2 × 2实非模矩阵。洛伦兹流形或“时空”是广义相对论的基本对象之一。其中一些物体具有许多对称性，而大多数物体根本没有对称性。修复一个具有大量对称性的时空。对于这个时空中的任何一点，如果我们按照所考虑的所有对称性移动它，那么得到的图像点的集合称为该点的“轨道”。许多研究人员已经注意到，用混沌轨道构造时空是很困难的，混沌轨道的意思是，广义地说，一些轨道反复返回（或几乎返回）同一地点。事实上，有许多方法可以精确地定义混沌这个词，这意味着已经得到了许多不同的定理。我打算沿着这条路线继续取得成果。

Proposal: DMS-0072165PI Scot AdamsThe study of Lorentz manifolds may be motivated by physicalconsiderations, since the Einstein field equations determine Lorentzmetrics on space-time. Following F.~Klein's Erlanger program, onemethod of approaching a geometric object is to study its group ofisometries, and it has long been known that the isometry group of aLorentz manifold is a Lie group. Moreover, any Lie group acts onitself (by left translation) preserving any left-invariant Lorentzmetric. However, as originally noticed by R.~Zimmer, M.~Gromov, andN.~Kowalsky, if one requires even a small amount of complication tothe dynamics, then the list of Lie groups admitting a Lorentz actionbecomes quite restricted. My work over the last few years has focusedon quantifying exactly which connected Lie groups admit a complicatedaction by isometries of a connected Lorentz manifold. As thedefinition of ``complicated'' varies, the answer varies, and I intendto look at a few more of these variations over the coming years. Inparticular, I would like to determine the collection of connected,noncompact, simple Lie groups admitting a locally faithful, nontame,isometric action on a connected Lorentz manifold. I believe it shouldbe true that any such group either has infinite center or is locallyisomorphic to the Lie group of 2 by 2 real unimodular matrices.A Lorentz manifold, or ``space-time'' is one of the basic object ofgeneral relativity. Some of these objects have many symmetries, whilemost have none at all. Fix a space-time having a large collection ofsymmetries. For any point in this space-time, if we move it around byall the symmetries under consideration, the collection of image pointsso obtained is called the ``orbit'' of the point. A number ofresearchers have noticed that it is difficult to construct aspace-time with chaotic orbits, by which we mean, broadly speaking,that some orbits repeatedly return (or nearly return) to thesame place. There are, in fact, many ways to define precisely the wordchaotic, and this means that a number of different theorems have beenobtained. I intend to continue developing results along these lines.