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Register a new userOn the harmonicity of normal almost contact metric structures
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We consider normal almost contact structures on a Riemannian manifold and, through their associated sections of an ad-hoc twistor bundle, study their harmonicity, as sections or as maps. We rewrite these harmonicity equations in terms of the Riemann curvature tensor and find conditions relating the harmonicity of the almost contact and almost complex structures of the total and base spaces of the Morimoto fibration.
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An almost contact metric structure is parametrized by a section of an associated homogeneous fibre bundle, and conditions for this to be a harmonic section, and a harmonic map, are studied. These involve the characteristic vector field, and the almost complex structure in the contact subbundle. Several examples are given where the harmonic section equations reduce to those for the characteristic field to be a harmonic section of the unit tangent bundle. These include trans-Sasakian structures, and certain nearly cosymplectic structures. On the other hand, we obtain examples where the characteristic field is harmonic but the almost contact structure is not. Many of our examples are obtained by considering hypersurfaces of almost Hermitian manifolds, with the induced almost contact structure, and comparing the harmonic section equations for both structures.
We study the geometry of almost contact pseudo-metric manifolds in terms of tensor fields $h:=frac{1}{2}pounds _xi varphi$ and $ell := R(cdot,xi)xi$, emphasizing analogies and differences with respect to the contact metric case. Certain identities involving $xi$-sectional curvatures are obtained. We establish necessary and sufficient condition for a nondegenerate almost $CR$ structure $(mathcal{H}(M), J, theta)$ corresponding to almost contact pseudo-metric manifold $M$ to be $CR$ manifold. Finally, we prove that a contact pseudo-metric manifold $(M,varphi,xi,eta,g)$ is Sasakian if and only if the corresponding nondegenerate almost $CR$ structure $(mathcal{H}(M), J)$ is integrable and $J$ is parallel along $xi$ with respect to the Bott partial connection.
Almost contact structures can be identified with sections of a twistor bundle and this allows to define their harmonicity, as sections or maps. We consider the class of nearly cosymplectic almost contact structures on a Riemannian manifold and prove curvature identities which imply the harmonicity of their parametrizing section, thus complementing earlier results on nearly-K{a}hler almost complex structures.
We study harmonic almost contact structures in the context of contact metric manifolds, and an analysis is carried out when such a manifold fibres over an almost Hermitian manifold, as exemplified by the Boothby-Wang fibration. Two types of almost contact metric warped products are also studied, relating their harmonicity to that of the almost Hermitian structure on the base or fibre.
We prove that the normal metric contact pairs with orthogonal characteristic foliations, which are either Bochner flat or locally conformally flat, are locally isometric to the Hopf manifolds. As a corollary we obtain the classification of locally conformally flat and Bochner-flat non-Kahler Vaisman manifolds.
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