Non-stationary smooth geometric structures for contracting measurable cocycles

We implement a differential-geometric approach to normal forms for contracting measurable cocycles to \mbox{Diff}^q({\bf R}^n, {\bf 0}), $q \geq2$. We obtain resonance polynomial normal forms for the contracting cocycle and its centralizer, via $C^q$ changes of coordinates. These are interpreted as nonstationary invariant differential-geometric structures. We also consider the case of contracted foliations in a manifold, and obtain $C^q$ homogeneous structures on leaves for an action of the group of subresonance polynomial diffeomorphisms together with translations

Quasihomogeneous three-dimensional real analytic Lorentz metrics do not exist

We show that a germ of a real analytic Lorentz metric on ${\bf R}^3$ which is locally homogeneous on an open set containing the origin in its closure is necessarily locally homogeneous. We classifiy Lie algebras that can act quasihomogeneously---meaning they act transitively on an open set admitting the origin in its closure, but not at the origin---and isometrically for such a metric. In the case that the isotropy at the origin of a quasihomogeneous action is semisimple, we provide a complete set of normal forms of the metric and the action.Comment: 23 pp. Took the place of "Quasihomogeneous three-dimensional real analytic Lorentz metrics" (arXiv:1401.6272), which was withdrawn by the first author. Revised version incorporates several minor corrections, including those suggested by the refere

Essential Killing fields of parabolic geometries

We study vector fields generating a local flow by automorphisms of a parabolic geometry with higher order fixed points. We develop general tools extending the techniques of [1], [2], and [3]. We apply these tools to almost Grassmannian, almost quaternionic, and contact parabolic geometries, including CR structures, to obtain descriptions of the possible dynamics of such flows near the fixed point and strong restrictions on the curvature. In some cases, we can show vanishing of the curvature on a nonempty open set. Deriving consequences for a specific geometry entails evaluating purely algebraic and representation-theoretic criteria in the model homogeneous space. Published in Indiana University Mathematics Journal.Comment: 50 pages. Minor corrections, references update

Conformal actions of nilpotent groups on pseudo-Riemannian manifolds

We study conformal actions of connected nilpotent Lie groups on compact pseudo-Riemannian manifolds. We prove that if a type-(p,q) compact manifold M supports a conformal action of a connected nilpotent group H, then the degree of nilpotence of H is at most 2p+1, assuming p <= q; further, if this maximal degree is attained, then M is conformally equivalent to the universal type-(p,q), compact, conformally flat space, up to finite covers. The proofs make use of the canonical Cartan geometry associated to a pseudo-Riemannian conformal structure.Comment: 41 pages, 3 figures. Article has been shortened from previous version, and several corrections have been made according to referees' suggestion

Strongly essential flows on irreducible parabolic geometries

We study the local geometry of irreducible parabolic geometries admitting strongly essential flows; these are flows by local automorphisms with higher-order fixed points. We prove several new rigidity results, and recover some old ones for projective and conformal structures, which show that in many cases the existence of a strongly essential flow implies local flatness of the geometry on an open set having the fixed point in its closure. For almost c-projective and almost quaternionic structures we can moreover show flatness of the geometry on a neighborhood of the fixed point.Comment: 34 pages. Proof of Proposition 3.1 significantly shortened, under slightly less general hypotheses (see Remark 3.1). Typos corrected and references updated. To appear in Transactions of the AM

C^1 Deformations of almost-Grassmannian structures with strongly essential symmetry

We construct a family of $(2,n)$-almost Grassmannian structures of regularity $C^1$, each admitting a one-parameter group of strongly essential automorphisms, and each not flat on any neighborhood of the higher-order fixed point. This shows that Theorem 1.3 of [9] does not hold assuming only $C^1$ regularity of the structure (see also [2, Prop 3.5]).Comment: 24 p