Topological sphere theorems for submanifolds and homology groups
Project no. 20776 · Status: Kapanmış · 2019
Summary
We show that three dimensional CR-warped product submanifold of nearly Kaehler six dimensional sphere is homotopic to a sphere under certain conditions. Homology of a contact CR-submanifold of a real hypersurface, which has naturally almost contact metric structure induced from the complex Euclidean space C^m ,is examined. More precisely, nonexistence of stable integral currents on a compact contact CR-submanifold of real hypersurface of canonical complex space form C^m is investigated and vanishing theorems concerning the homology groups are obtained. We investigate topological sphere theorems for compact minimal contact CR-submanifolds of odd dimensional unit sphere. We show that if an inequality involving the warping function and the scalar curvature of the fibers is satisfied, a compact minimal contact CR-warped product submanifold of the odd dimensional unit sphere is homeomorphic to the sphere. In particular case, for 5- dimensional unit sphere, we show that a 4- dimensional compact minimal contact CR-warped product submanifold is homeomorphic to a sphere if ??lnf?^2<1 is satisfied. By using Bonnet-Myers's theorem we give a result about fundamental group and by Leung's theorem we obtain a result about homology groups of a contact CR-warped submanifold.