David E. Blair

Bio: David E. Blair is an academic researcher from Michigan State University. The author has contributed to research in topics: Ricci-flat manifold & Metric (mathematics). The author has an hindex of 23, co-authored 81 publications receiving 4978 citations. Previous affiliations of David E. Blair include University of Liverpool & University of Ioannina.

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

Corrected energy of distributions for 3-Sasakian and normal complex contact manifolds

Abstract: In this paper we show that the natural fibrations on 3-Sasakian manifolds and on normal complex contact metric manifolds are minima of the corrected energy of the corresponding distributions.

A Survey of Riemannian Contact Geometry

Abstract: Abstract This survey is a presentation of the five lectures on Riemannian contact geometry that the author gave at the conference “RIEMain in Contact”, 18-22 June 2018 in Cagliari, Sardinia. The author was particularly pleased to be asked to give this presentation and appreciated the organizers’ kindness in dedicating the conference to him. Georges Reeb once made the comment that the mere existence of a contact form on a manifold should in some sense “tighten up” the manifold. The statement seemed quite pertinent for a conference that brought together both geometers and topologists working on contact manifolds, whether in terms of “tight” vs. “overtwisted” or whether an associated metric should have some positive curvature. The first section will lay down the basic definitions and examples of the subject of contact metric manifolds. The second section will be a continuation of the first discussing tangent sphere bundles, contact structures on 3-dimensional Lie groups and a brief treatment of submanifolds. Section III will be devoted to the curvature of contact metric manifolds. Section IV will discuss complex contact manifolds and some older style topology. Section V treats curvature functionals and Ricci solitons. A sixth section has been added giving a discussion of the question of whether a Riemannian metric g can be an associated metric for more than one contact structure; at the conference this was an addendum to the third lecture.

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

The contact Whitney sphere

Abstract: In this paper, we introduce the contact Whitney sphere as an imbedding of the --dimensional unit sphere as an integral submanifold of the standard contact structure on . We obtain a general inequality for integral submanifolds in , involving both the scalar curvature and the mean curvature, and we use the equality case in order to characterize the contact Whitney sphere. We also study a similar problem foranti-invariant submanifolds of , tangent to the structure vector field.

A class of almost contact riemannian manifolds

Abstract: Recently S. Tanno has classified connected almostcontact Riemannian manifolds whose automorphism groups have themaximum dimension [9]. In his classification table the almost contactRiemannian manifolds are divided into three classes: (1) homogeneousnormal contact Riemannian manifolds with constant 0-holomorphic sec-tional curvature if the sectional curvature for 2-planes which contain

Real Killing spinors and holonomy

Abstract: We give a description of all complete simply connected Riemannian manifolds carrying real Killing spinors. Furthermore, we present a construction method for manifolds with the exceptional holonomy groupsG 2 and Spin(7).

G-Structures and Wrapped NS5-Branes

Abstract: We analyse the geometrical structure of supersymmetric solutions of type II supergravity of the form Open image in new window1,9−n×Mn with non-trivial NS flux and dilaton. Solutions of this type arise naturally as the near-horizon limits of wrapped NS fivebrane geometries. We concentrate on the case d=7, preserving two or four supersymmetries, corresponding to branes wrapped on associative or SLAG three-cycles. Given the existence of Killing spinors, we show that M7 admits a G2-structure or an SU(3)-structure, respectively, of specific type. We also prove the converse result. We use the existence of these geometric structures as a new technique to derive some known and new explicit solutions, as well as a simple theorem implying that we have vanishing NS three-form and constant dilaton whenever M7 is compact with no boundary. The analysis extends simply to other type II examples and also to type I supergravity.