Foliated Rigidity and Dynamics of Space-Time

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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.