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

/pdf/a-class-of-almost-contact-riemannian-manifolds-1yxyggdui1.pdf

A class of almost contact riemannian manifolds

Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

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).

https://projecteuclid.org/download/pdf_1/euclid.cmp/1104253076

Real Killing spinors and holonomy

http://www.dcs.gla.ac.uk/~muffy/papers/calder-kolberg-magill-reiff.pdf

Feature interaction: a critical review and considered forecast

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.

G-Structures and Wrapped NS5-Branes

/pdf/a-classification-of-3-dimensional-contact-metric-manifolds-2m6h4oa025.pdf

A classification of $3$-dimensional contact metric manifolds with $Q\phi=\phi Q$

It is first observed that on a 3-dimensional Sasakian manifold the torsion of a Legendre curve is identically equal to +1. It is then shown that, conversely, if a curve on a Sasakian 3-manifold has constant torsion +1 and satisfies the initial conditions at one point for a Legendre curve, it is a Legendre curve. Furthermore, among contact metric structures, this property is characteristic of Sasakian metrics. For the standard contact structure onR3 with its standard Sasakian metric the curvature of a Legendre curve is shown to be twice the curvature of its projection to thexy-plane with respect to the Euclidean metric. Thus this metric onR3 is more natural for the study of Legendre curves than the Euclidean metric.

On Legendre curves in contact 3-manifolds

for a given value of p. The form η defines a differential system and it is important to study the local and global properties of its integral manifolds. To this end, the notion of a quasi-Sasakian structure on an almost contact metric manifold was introduced by one of the authors [1] and its main properties developed. In the present paper their topological properties are considered and it is shown that both compact Sasakian and cosymplectic manifolds have global properties similar to compact Kaehler manifolds. Examples are the unit hypersphere S in Euclidean space, and in fact, the circle bundles over any compact Hodge variety. In the latter class, examples are provided by M X S where M is any compact Kaehler manifold. As one might expect, therefore, not only locally, but topologically as well, the compact cosymplectic spaces are the proper odd dimensional analogues of the compact Kaehler manifolds. A complete, but not compact, simply connected cosymplectic manifold is a product with one factor Kaehlerian. The notation and terminology in this paper will be the same as that employed in [1].

/pdf/topology-of-almost-contact-manifolds-4r2nhlkbkf.pdf

Topology of almost contact manifolds

Almost contact manifolds

Under an assumption of regularity a manifold with an f-structure satisfying certain conditions analogous to those of a Kahler structure admits a fibration a s a principal toroidal bundle over a Kaller manifold. In some natural special cases, additional information about the bundle spaceis obtained. Finally, curvature relations between the bundle space and the base space are studied.

/pdf/differential-geometric-structures-on-principal-toroidal-1awx8za6gr.pdf

David E. Blair

Papers

A classification of $3$-dimensional contact metric manifolds with $Q\phi=\phi Q$

On Legendre curves in contact 3-manifolds

Topology of almost contact manifolds

Almost contact manifolds

Differential geometric structures on principal toroidal bundles