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Register a new userOn the curvature of metric contact pairs
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We consider manifolds endowed with metric contact pairs for which the two characteristic foliations are orthogonal. We give some properties of the curvature tensor and in particular a formula for the Ricci curvature in the direction of the sum of the two Reeb vector fields. This shows that metrics associated to normal contact pairs cannot be flat. Therefore flat non-Kahler Vaisman manifolds do not exist. Furthermore we give a local classification of metric contact pair manifolds whose curvature vanishes on the vertical subbundle. As a corollary we have that flat associated metrics can only exist if the leaves of the characteristic foliations are at most three-dimensional.
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A contact pair on a manifold always admits an associated metric for which the two characteristic contact foliations are orthogonal. We show that all these metrics have the same volume element. We also prove that the leaves of the characteristic foliations are minimal with respect to these metrics. We give an example where these leaves are not totally geodesic submanifolds.
We show that $phi$-invariant submanifolds of metric contact pairs with orthogonal characteristic foliations make constant angles with the Reeb vector fields. Our main result is that for the normal case such submanifolds of dimension at least $2$ are all minimal. We prove that an odd-dimensional $phi$-invariant submanifold of a metric contact pair with orthogonal characteristic foliations inherits a contact form with an almost contact metric structure, and this induced structure is contact metric if and only if the submanifold is tangent to one Reeb vector field and orthogonal to the other one. Furthermore we show that the leaves of the two characteristic foliations of the differentials of the contact pair are minimal. We also prove that when one Reeb vector field is Killing and spans one characteristic foliation, the metric contact pair is a product of a contact metric manifold with $mathbb{R}$.
We prove that the universal covering of a complete locally symmetric normal metric contact pair manifold is a Calabi-Eckmann manifold. Moreover we show that a complete, simply connected, normal metric contact pair manifold such that the foliation induced by the vertical subbundle is regular and reflections in the integral submanifolds of the vertical subbundle are isometries, then the manifold is the product of globally $phi$-symmetric spaces and fibers over a locally symmetric space endowed with a symplectic pair.
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.
We consider manifolds endowed with a contact pair structure. To such a structure are naturally associated two almost complex structures. If they are both integrable, we call the structure a normal contact pair. We generalize the Morimotos Theorem on product of almost contact manifolds to flat bundles. We construct some examples on Boothby--Wang fibrations over contact-symplectic manifolds. In particular, these results give new methods to construct complex manifolds.
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