arXiv subjects

Amine Hadjar

Publications and source records attributed to Amine Hadjar.

6 recordsLinked to original sources

arXiv2026

On characteristic foliations of metric contact-symplectic structures

We study compatible and associated metrics for a contact-symplectic pair $(\eta , \omega)$ on a manifold. We show that the integral curves of the Reeb vector field are geodesics for any compatible metric. We prove that all associated metrics share a common volume element, which we give explicitly. When the characteristic foliations of $\eta$ and $\omega$ are orthogonal with respect to an associated metric, their leaves, as well as those of the characteristic foliation of $d\eta$, are minimal. We construct explicit examples on nilpotent Lie groups and nilmanifolds where the characteristic foliations are not both totally geodesic.

math.DG↗

arXiv2015

Invariant submanifolds of metric contact pairs

We show that $ϕ$-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 $ϕ$-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 $\mathbb{R}$.

math.DG↗

arXiv2015

Bochner and Conformal Flatness of Normal Metric Contact Pairs

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-Kähler Vaisman manifolds.

math.DG↗

arXiv2015

Minimality of invariant submanifolds in Metric Contact Pair Geometry

We study invariant submanifolds of manifolds endowed with a normal or complex metric contact pair with decomposable endomorphism field $ϕ$. For the normal case, we prove that a $ϕ$-invariant submanifold tangent to a Reeb vector field and orthogonal to the other one is minimal. For a $ϕ$-invariant submanifold $N$ everywhere transverse to both the Reeb vector fields but not orthogonal to them, we prove that it is minimal if and only if the angle between the tangential component $ξ$ (with respect to $N$) of a Reeb vector field and the Reeb vector field itself is constant along the integral curves of $ξ$. For the complex case (when just one of the two natural almost complex structures is supposed to be integrable), we prove that a complex submanifold is minimal if and only if it is tangent to both the Reeb vector fields.

math.DG↗

arXiv2010

On the Characteristic Foliations of Metric Contact Pairs

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.

math.DG↗

arXiv2008

Contact Pairs

We introduce a new geometric structure on differentiable manifolds. A \textit{Contact} \textit{Pair}on a manifold $M$ is a pair $(α,η) $ of Pfaffian forms of constant classes $2k+1$ and $2h+1$ respectively such that $α\wedge dα^{k}\wedgeη\wedge dη^{h}$ is a volume form. Both forms have a characteristic foliation whose leaves are contact manifolds. These foliations are transverse and complementary. Further differential objects are associated to Contact Pairs: two commuting Reeb vector fields, Legendrian curves on $M$ and two Lie brackets on $\mathcal{C}^{\infty}(M) $. We give a local model and several existence theorems on nilpotent Lie groups, nilmanifolds, bundles over the circle and principal torus bundles.

math.DG↗