Mathematics – Differential Geometry
Scientific paper
2005-05-30
International Journal of Mathematics Vol. 17, No. 1 (2006) 119-227
Mathematics
Differential Geometry
8 pages
Scientific paper
A. Borel proved that, if a finite group $F$ acts effectively and continuously on a closed aspherical manifold $M$ with centerless fundamental group $\pi_1(M)$, then a natural homomorphism $\psi$ from $F$ to the outer automorphism group ${\rm Out} \pi_1(M)$ of $\pi_1(M)$, 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 $\pi_1(N)$ has a non-trivial center. It turns out that if $G$ attains the maximal dimension equal to the rank of Center $\pi_1(N)$ 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 $\pi_1(N)$ is torsion free, then the associated abstract kernel $\psi: G/G^0\to {\rm Out} \pi_1(N)$ 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.
No associations
LandOfFree
Compact Group Actions On Closed Manifolds of Non-positive Curvature does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Compact Group Actions On Closed Manifolds of Non-positive Curvature, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Compact Group Actions On Closed Manifolds of Non-positive Curvature will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-314936