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Rigid geometric structures, isometric actions, and algebraic quotients

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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.

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Authors and Affiliations

  1. LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Jinpeng An

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  1. Jinpeng An
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Correspondence to Jinpeng An.

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Research partially supported by NSFC grant 10901005 and FANEDD grant 200915.

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An, J. Rigid geometric structures, isometric actions, and algebraic quotients. Geom Dedicata 157, 153–185 (2012). https://doi.org/10.1007/s10711-011-9603-2

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