2010/05/09 by Jinpeng An, An, Jinpeng
Mathematics · #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1005.1423
openalex publication_date 2010/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \widetildeM 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 phenomena that if G is split solvable and large then π1(M) is also large, and (3) two fixed point theorems for split solvable G and compact analytic M with non-unimodular σ.