Showing papers by "David E. Blair published in 2011"

Bochner and conformal flatness on normal complex contact metric manifolds

Abstract: We will prove that normal complex contact metric manifolds that are Bochner flat must have constant holomorphic sectional curvature 4 and be Kahler. If they are also complete and simply connected, they must be isometric to the odd-dimensional complex projective space $${{\mathbb{C}P^{2n+1}}}$$ (4) with the Fubini-Study metric. On the other hand, it is not possible for normal complex contact metric manifolds to be conformally flat.

On the curvature of metric contact pairs

Abstract: We consider manifolds endowed with metric contact pairs for which the two characteristic foliations are orthogonal. We give some properties of the curvature tensor and in particular a formula for the Ricci curvature in the direction of the sum of the two Reeb vector fields. This shows that metrics associated to normal contact pairs cannot be flat. Therefore flat non-K\"ahler Vaisman manifolds do not exist. Furthermore we give a local classification of metric contact pair manifolds whose curvature vanishes on the vertical subbundle. As a corollary we have that flat associated metrics can only exist if the leaves of the characteristic foliations are at most three-dimensional.

Symmetry in the Geometry of Metric Contact Pairs

Abstract: We prove that the universal covering of a complete locally symmetric normal metric contact pair manifold is a Calabi-Eckmann manifold. Moreover we show that a complete, simply connected, normal metric contact pair manifold such that the foliation induced by the vertical subbundle is regular and reflections in the integral submanifolds of the vertical subbundle are isometries, then the manifold is the product of globally $\phi$-symmetric spaces and fibers over a locally symmetric space endowed with a symplectic pair.