f-Biharmonic Curves in the Three-dimensional Para-Sasakian Space Forms

In this paper, we give some characterizations for proper f-biharmonic curves in the para-Bianchi-Cartan-Vranceanu space forms with 3-dimensional para-Sasakian structures.

On the classification of Smale–Barden manifolds with Sasakian structures

Smale–Barden manifolds [Formula: see text] are classified by their second homology [Formula: see text] and the Barden invariant [Formula: see text]. It is an important and difficult question to decide when [Formula: see text] admits a Sasakian structure in terms of these data. In this work, we show methods of doing this. In particular, we realize all [Formula: see text] with [Formula: see text] and [Formula: see text] provided that [Formula: see text], [Formula: see text], [Formula: see text] are pairwise coprime. We give a complete solution to the problem of the existence of Sasakian structures on rational homology spheres in the class of semi-regular Sasakian structures. Our method allows us to completely solve the following problem of Boyer and Galicki in the class of semi-regular Sasakian structures: determine which simply connected rational homology 5-spheres admit negative Sasakian structures.

Sasakians and the geometry of a mass term

A gauge-invariant mass term for nonabelian gauge fields in two dimensions can be expressed as the Wess–Zumino–Witten (WZW) action. Hard thermal loops in the gauge theory in four dimensions at finite temperatures generate a screening mass for some components of the gauge field. This can be expressed in terms of the WZW action using the bundle of complex structures (for Euclidean signature) or the bundle of lightcones over Minkowski space. We show that a dynamically generated mass term in three dimensions can be put within the same general framework using the bundle of Sasakian structures.

Almost contact Hom-Lie algebras and Sasakian Hom-Lie algebras

In this paper, we consider Hom-Lie groups and introduce left invariant almost contact structures on them (almost contact Hom-Lie algebras). On such Hom-Lie groups, we construct the almost contact metrics and the contact forms. We give the notion of normal almost contact Hom-Lie algebras and describe [Formula: see text]-contact and Sasakian structures on Hom-Lie algebras. Also, we study some of their properties. In addition, it is shown that any Sasakian Hom-Lie algebra is a [Formula: see text]-contact Hom-Lie algebra. Finally, we present examples of Sasakian Hom-Lie algebras and in particular, we show that the skew symmetric matrix [Formula: see text] carries a Sasakian structure.

Transverse Kähler–Ricci Solitons of Five-Dimensional Sasaki–Einstein Spaces Yp,q and T1,1

We investigate the deformations of the Sasaki–Einstein structures of the five-dimensional spaces T 1 , 1 and Y p , q by exploiting the transverse structure of the Sasaki manifolds. We consider local deformations of the Sasaki structures preserving the Reeb vector fields but modify the contact forms. In this class of deformations, we analyze the transverse Kähler–Ricci flow equations. We produce some particular explicit solutions representing families of new Sasakian structures.

Contact Structures of Sasaki Type and Their Associated Moduli

This article is based on a talk at the RIEMain in Contact conference in Cagliari, Italy in honor of the 78th birthday of David Blair one of the founders of modern Riemannian contact geometry. The present article is a survey of a special type of Riemannian contact structure known as Sasakian geometry. An ultimate goal of this survey is to understand the moduli of classes of Sasakian structures as well as the moduli of extremal and constant scalar curvature Sasaki metrics, and in particular the moduli of Sasaki-Einstein metrics.

Homology Smale–Barden Manifolds with K-contact but not Sasakian Structures

Kollár has found subtle obstructions to the existence of Sasakian structures on five-dimensional manifolds. In the present article we develop methods of using these obstructions to distinguish K-contact manifolds from Sasakian ones. In particular, we find the first example of a closed $5$-manifold $M$ with $H_1(M,\mathbb{Z})=0,$ which is K-contact but which carries no semi-regular Sasakian structures.

Sasakian structures a foliated approach

Recent renewed interest in Sasakian manifolds is due mainly to the fact that they can provide examples of generalized Einstein manifolds, manifolds which are of great interest in mathematical models of various aspects of physical phenomena. Sasakian manifolds are odd dimensional counterparts of Kählerian manifolds to which they are closely related. The paper presents a foliated approach to Sasakian manifolds on which the author gave several lectures. The paper concentrates on cohomological properties of Sasakian manifolds and of transversely holomorphic and Kählerian foliations. These properties permit to formulate obstructions to the existence of Sasakian structures on compact manifolds.