Domenico Perrone

Homogeneity on three-dimensional contact metric manifolds

We study ball-homogeneity, curvature homogeneity, natural reductivity, conformal flatness and ϕ-symmetry for three-dimensional contact metric manifolds. Several classification results are given.

Unit Tangent Sphere Bundles and Two-Point Homogeneous Spaces

We characterize two-point homogeneous spaces, locally symmetric spaces, C and B-spaces via properties of the standard contact metric structure of their unit tangent sphere bundle. Further, under various conditions on a Riemannian manifold, we show that its unit tangent sphere bundle is a (locally) homogeneous contact metric space if and only if the manifold itself is (locally) isometric to a two-point homogeneous space.

Harmonic Vector Fields

Publisher Summary This chapter discusses fundamental matters such as the Dirichlet energy and tension tensor of a unit tangent vector field on a Riemannian manifold, first and second variation formulae, and the harmonic vector fields system. The study of the weak solutions to this system (existence and local properties) is missing from the present day mathematical literature. Various instances are investigated where harmonic vector fields occur and to generalizations. Any unit vector field that is a harmonic map is also a harmonic vector field. The study of harmonic map system is more appropriate on a Hermitian manifold and that results in Hermitian harmonic maps to be useful in studying rigidity of complete Hermitian manifolds.

MINIMALITY, HARMONICITY AND CR GEOMETRY FOR REEB VECTOR FIELDS

Let (M, g) be a Riemannian manifold and T1 M its unit tangent sphere bundle. Minimality and harmonicity of unit vector fields have been extensively studied by considering on T1M the Sasaki metric . This metric, and other well-known Riemannian metrics on T1 M, are particular examples of Riemannian natural metrics. In this paper we equip T1 M with a Riemannian natural metric and in particular with a natural contact metric structure. Then, we study the minimality for Reeb vector fields of contact metric manifolds and of quasi-umbilical hypersurfaces of a Kahler manifold. Several explicit examples are given. In particular, the Reeb vector field ξ of a K-contact manifold is minimal for any that belongs to a family depending on two parameters of metrics of the Kaluza–Klein type. Next, we show that the Reeb vector field ξ of a K-contact manifold defines a harmonic map for any Riemannian natural metric . Besides this, if the Reeb vector ξ of an almost contact metric manifold is a CR map then the induced almost CR structure on M is strictly pseudoconvex and ξ is a pseudo-Hermitian map; if in addition ξ is geodesic then is a harmonic map. Moreover, the Reeb vector field ξ of a contact metric manifold is a CR map iff ξ is Killing and is a special metric of the Kaluza–Klein type. Finally, in the final section, we obtain that there is a family of strictly pseudoconvex CR structures on T1S2n+1 depending on one parameter, for which a Hopf vector field ξ determines a pseudo-harmonic map (in the sense of Barletta–Dragomir–Urakawa [8]) from S2n+1 to T1S2n+1.

Torsion tensor and critical metrics on contact (2n+1)-manifolds

Contact Riemannian manifolds (M, Ω,g) satisfying the condition (1) ∇ξτ=0, where τ is the torsion introduced byChern andHamilton [6] and ξ is the characteristic vector field, have interesting geometric properties (see [6], [9], [11]). In this paper we give a variational characterization of compact contact Riemannian manifolds which satisfy (1). Moreover we study the tangent sphere bundles (T1M, ω, g), where (ω,g) is the standard contact Riemannian structure, which satisfy the condition (1); in particular in the 3-dimensional case we find a surprising result (see Corollary 5.3).

Torsion and homogeneity on contact metric three-manifolds

We show that a three-dimensional contact metric manifold is locally homogeneous if and only if it is ball-homogeneous and satisfies the condition ∇ξτ=2aτϕ, with a constant. Then, we relate the condition ∇ξτ=0 with the existence of taut contact circles on a compact three-dimensional contact metric manifold.

On the minimal eigenvalue of the Laplacian operator for

Let (M, g) be a compact orientable conformally flat Riemannian manifold and pXi the minimal eigenvalue of the Laplacian operator for p-forms. We prove that if there exists a positive constant K such that p > Kg, where p is the Ricci tensor of M, then pXi > Kp(n — p + l)/(n — 1) f°r eacn P, 1 < p < n/2, (n = dim M); moreover if the equality holds for some p then M is of constant curvature a = K/(n — 1).

p

In this paper we characterize K-contact semi-Riemannian manifolds and Sasakian semi-Riemannian manifolds in terms of curvature. Moreover, we show that any conformally flat K-contact semi-Riemannian manifold is Sasakian and of constant sectional curvature κ = ε, where ε = ±1 denotes the causal character of the Reeb vector field. Finally, we give some results about the curvature of a K-contact Lorentzian manifold.

-forms in conformally flat Riemannian manifolds

The main result of this paper gives a characterization of complex 2-dimensional cohomological Einstein Kaehler manifolds. As a consequence we obtain some interesting restrictions for the constructions of cohomological Einstein metrics on complex 2-manifolds.

Curvature of K -contact Semi-Riemannian Manifolds

Abstract. In a recent paper [10] we introduced the notion of Levi har-monic map f from an almost contact semi-Riemannian manifold (M,ϕ,ξ,η,g) into a semi-Riemannian manifold M ′ . In particular, we computedthe tension field τ H (f) for a CR map f between two almost contactsemi-Riemannian manifolds satisfying the so-called ϕ-condition, whereH= Ker(η) is the Levi distribution. In the present paper we show thatthe condition (A) of Rawnsley [17] is related to the ϕ-condition. Then,we compute the tension field τ H (f) for a CR map between two arbitraryalmost contact semi-Riemannian manifolds, and we study the conceptof Levi pluriharmonicity. Moreover, we study the harmonicity on quasi-cosymplectic manifolds. 1. IntroductionAs a natural continuation of the ideas in [2], and following the ideas of B. Fu-glede (who started the study of the semi-Riemannian case within harmonic maptheory, cf. [11] and [1], pp. 427–455), in the recent paper [10] S. Dragomir andthe present author introduced the concept of Levi harmonic map f from an al-most contact semi-Riemannian manifold (M,ϕ,ξ,η,g) into a semi-Riemannianmanifold (M