2003/10/06 by Su-Jen Kan, Kan, Su-Jen
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.CV #msc:32C09 #msc:32Q28 #msc:32Q45
paper · pdf · doi:10.48550/arxiv.math/0310069
23 pages. J.reine angew. Math. (to appear)
arxiv created 2004/03/10 · arxiv updated 2009/12/01
Given a real-analytic Riemannian manifold X there is a canonical complex structure, which is compatible with the canonical complex structure on T^*X and makes the leaves of the Riemannian foliation on TX into holomorphic curves, on its tangent bundle. A \it Grauert tube over X of radius r, denoted as TrX, is the collection of tangent vectors of X of length less than r equipped with this canonical complex structure. In this article, we prove the following two rigidity property of Grauert tubes. First, for any real-analytic Riemannian manifold such that rmax>0, we show that the identity component of the automorphism group of TrX is isomorphic to the identity component of the isometry group of X provided that r<rmax. Secondly, let X be a homogeneous Riemannian manifold and let the radius r<rmax, then the automorphism group of TrX is isomorphic to the isometry group of X and there is a unique Grauert tube representation for such a complex manifold TrX.