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We study invariant submanifolds of manifolds endowed with a normal or complex metric contact pair with decomposable endomorphism field $phi$. For the normal case, we prove that a $phi$-invariant submanifold tangent to a Reeb vector field and orthogonal to the other one is minimal. For a $phi$-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 $xi$ (with respect to $N$) of a Reeb vector field and the Reeb vector field itself is constant along the integral curves of $xi$. 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.
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
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 ind
Unlike Legendrian submanifolds, the deformation problem of coisotropic submanifolds can be obstructed. Starting from this observation, we single out in the contact setting the special class of integral coisotropic submanifolds as the direct generaliz
In $N(k)$-contact metric manifolds and/or $(k,mu)$-manifolds, gradient Ricci solitons, compact Ricci solitons and Ricci solitons with $V$ pointwise collinear with the structure vector field $xi $ are studied.
Given a Riemannian manifold $N^n$ and ${cal Z}in mathfrak{X}(N)$, an isometric immersion $fcolon M^mto N^n$ is said to have the emph{constant ratio property with respect to ${cal Z}$} either if the tangent component ${cal Z}^T_f$ of ${cal Z}$ vanishe