Showing papers by "David E. Blair published in 2013"

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-Kahler 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

D-homothetic warping

Abstract: The goal of this lecture will be to introduce the notion ofD-homothetic warping, give a few rudimentary properties and a couple of applications. As with the usual warped product it is hoped that this idea will prove useful for generating further results and examples of various structures. Details of the proofs will appear in [3]. For this purpose we must first review the geometry of contact metric and almost contact metric manifolds. By a contact manifold we mean a C manifold M2n+1 together with a 1-form η such that η ∧ (dη) 6= 0. It is well known that given η there exists a unique vector field ξ such that dη(ξ,X) = 0 and η(ξ) = 1. The vector field ξ is known as the characteristic vector field or Reeb vector field of the contact structure η. Denote by D the contact subbundle defined by {X ∈ TmM : η(X) = 0}. A Riemannian metric g is an associated metric for a contact form η if, first of all, η(X) = g(X, ξ) and secondly, there exists a field of endomorphisms, φ, such that φ2 = −I + η ⊗ ξ, dη(X,Y ) = g(X,φY ). We refer to (φ, ξ, η, g) as a contact metric structure and to M2n+1 with such a structure as a contact metric manifold. By an almost contact manifold we mean a C manifold M2n+1 together with a field of endomorphisms φ, a 1-form η and a vector field ξ such that

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 with decomposable ϕ is a Calabi-Eckmann manifold or the Riemannian product of a sphere and . We show that a complete, simply connected, normal metric contact pair manifold with decomposable ϕ, such that the foliation induced by the vertical subbundle is regular and reflections in the integral submanifolds of the vertical subbundle are isometries, is the product of globally ϕ-symmetric spaces or the product of a globally ϕ-symmetric space and . Moreover in the first case the manifold fibers over a locally symmetric space endowed with a symplectic pair.