This paper presents a study of contact metric manifolds for which the characteristic vector field of the contact structure satisfies a nullity type condition, condition (*) below. There are a number of reasons for studying this condition and results concerning it given in the paper: There exist examples in all dimensions; the condition is invariant underD-homothetic deformations; in dimensions>5 the condition determines the curvature completely; and in dimension 3 a complete, classification is given, in particular these include the 3-dimensional unimodular Lie groups with a left invariant metric.

Contact metric manifolds satisfying a nullity condition

The canonical paracontact connection is defined and it is shown that its torsion is the obstruction the paracontact manifold to be paraSasakian. A $${\mathcal{D}}$$
 -homothetic transformation is determined as a special gauge transformation. The η-Einstein manifold are defined, it is proved that their scalar curvature is a constant, and it is shown that in the paraSasakian case these spaces can be obtained from Einstein paraSasakian manifolds with $${\mathcal{D}}$$
 -homothetic transformations. It is shown that an almost paracontact structure admits a connection with totally skew-symmetric torsion if and only if the Nijenhuis tensor of the paracontact structure is skew-symmetric and the defining vector field is Killing.

Canonical connections on paracontact manifolds

Generalized Sasakian-space-forms are introduced and studied. Many examples of these manifolds are presented, by using some different geometric techniques such as Riemannian submersions, warped products or conformal and related transformations. New results on generalized complex-space-forms are also obtained.

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Generalized Sasakian-space-forms

Inspired by a result of Boyer and Galicki, we prove that a complete K-contact gradient soliton is compact Einstein and Sasakian. For the non-gradient case we show that the soliton vector field is a Jacobi vector field along the geodesics of the Reeb vector field. Next we show that among all complete and simply connected K-contact manifolds only the unit sphere admits a non-Killing holomorphically planar conformal vector field (HPCV). Finally we show that, if a (k, μ)-contact manifold admits a non-zero HPCV, then it is either Sasakian or locally isometric to E3 or En+1 × Sn (4).

Certain Results on K-Contact and (k, μ)-Contact Manifolds

We consider locally symmetric almost Kenmotsu manifolds showing that such a manifold is a Kenmotsu manifold if and only if the Lie derivative of the structure, with respect to the Reeb vector field $\xi$, vanishes. Furthermore, assuming that for a $(2n+1)$-dimensional locally symmetric almost Kenmotsu manifold such Lie derivative does not vanish and the curvature satisfies $R_{XY}\xi =0$ for any $X, Y$ orthogonal to $\xi$, we prove that the manifold is locally isometric to the Riemannian product of an $(n+1)$-dimensional manifold of constant curvature $-4$ and a flat $n$-dimensional manifold. We give an example of such a manifold.

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Almost Kenmotsu manifolds and local symmetry

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

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

A new class of 3-dimensional contact metric manifolds is found. Moreover it is proved that there are no such manifolds in dimensions greater than 3.

On the Existence of a New Class of Contact Metric Manifolds

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The harmonicity of the Reeb vector field on contact metric 3-manifolds

In this paper we study contact metric manifoldsM2n+1(ϕ, η, ξ,g) with characteristic vector field ξ belonging to thek-nullity distribution. Moreover we prove that there exist i) nonK-contact, contact metric manifolds of dimension greater than 3 with Ricci operator commuting with ϕ and ii) 3-dimensional contact metric manifolds with non-zero constant ϕ-sectional curvature.

Themis Koufogiorgos

Papers

Contact metric manifolds satisfying a nullity condition

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

On the Existence of a New Class of Contact Metric Manifolds

The harmonicity of the Reeb vector field on contact metric 3-manifolds

Contact metric manifolds