5-dimensional contact manifolds with second Betti number \(b_ 2=0\) (Q1108597)

Let M be a 5-dimensional simply-connected compact regular Sasakian manifold with \(b_ 2=0\). If the scalar curvature \(r>-4\) (resp. \(r=const.>-4)\), then M is homeomorphic (resp. isometric) to a sphere S 5. A conseqence of these results is the following: a 5-dimensional compact simply-connected regular Sasakian manifold either of positive curvature or \(\mu\)-holomorphically pinched with \(\mu >1/2\), is homeomorphic to a sphere S 5.