Compact Group Actions On Closed Manifolds of Non-positive Curvature

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dc.creatorXu, Bin
dc.date2005-05-30
dc.date.accessioned2026-07-07T13:12:49Z
dc.date.available2026-07-07T13:12:49Z
dc.descriptionA. Borel proved that, if a finite group $F$ acts effectively and continuously on a closed aspherical manifold $M$ with centerless fundamental group $π_1(M)$, then a natural homomorphism $ψ$ from $F$ to the outer automorphism group ${\rm Out} π_1(M)$ of $π_1(M)$, called the associated abstract kernel, is a monomorphism. In this paper, we investigate to what extent Borel's theorem holds for a compact Lie group $G$ acting effectively and smoothly on a particular orientable aspherical manifold $N$ admitting a Riemannian metric $g_0$ of non-positive curvature in case that $π_1(N)$ has a non-trivial center. It turns out that if $G$ attains the maximal dimension equal to the rank of Center $π_1(N)$ and the metric $g_0$ is real analytic, then any element of $G$ defining a diffemorphism homotopic to the identity of $N$ must be contained in the identity component $G^0$ of $G$. Moreover, if the inner automorphism group of $π_1(N)$ is torsion free, then the associated abstract kernel $ψ: G/G^0\to {\rm Out} π_1(N)$ is a monomorphism. The same result holds for the non-orientable $N$'s under certain techical assumptions. Our result is an application of a theorem by Schoen-Yau (Topology, {\bf 18} (1979), 361-380) on harmonic mappings.
dc.description8 pages
dc.identifierhttps://arxiv.org/abs/math/0505644
dc.identifierhttp://arxiv.org/abs/math/0505644
dc.identifierInternational Journal of Mathematics Vol. 17, No. 1 (2006) 119-227
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/229692
dc.subjectDifferential Geometry
dc.subjectGeometric Topology
dc.subject57S25, 53C43 (Primary); 20F34 (Secondary)
dc.titleCompact Group Actions On Closed Manifolds of Non-positive Curvature
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