Geometriae Dedicata

, Volume 157, Issue 1, pp 153–185 | Cite as

Rigid geometric structures, isometric actions, and algebraic quotients

Original Paper

Abstract

By using a Borel density theorem for algebraic quotients, we prove a theorem concerning isometric actions of a Lie group G on a smooth or analytic manifold M with a rigid A-structure σ. It generalizes Gromov’s centralizer and representation theorems to the case where R(G) is split solvable and G/R(G) has no compact factors, strengthens a special case of Gromov’s open dense orbit theorem, and implies that for smooth M and simple G, if Gromov’s representation theorem does not hold, then the local Killing fields on \({\widetilde{M}}\) are highly non-extendable. As applications of the generalized centralizer and representation theorems, we prove (1) a structural property of Iso(M) for simply connected compact analytic M with unimodular σ, (2) three results illustrating the phenomenon that if G is split solvable and large then π 1(M) is also large, and (3) two fixed point theorems for split solvable G and compact analytic M with non-unimodular σ.

Keywords

Rigid geometric structure Isometric action Invariant measure Fundamental group Algebraic quotient 

Mathematics Subject Classification (2000)

53C15 57S20 37B05 37C85 20G20 22D40 

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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.LMAM, School of Mathematical SciencesPeking UniversityBeijingChina

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