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Register a new userMinimality of invariant submanifolds in Metric Contact Pair Geometry
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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.
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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.
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 generalization of Legendrian submanifolds for what concerns deformation and moduli theory. Indeed, being integral coisotropic is proved to be a rigid condition, and moreover the integral coisotropic deformation problem is unobstructed with discrete moduli space.
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 ${cal Z}$ vanishes identically or if ${cal Z}^T_f$ vanishes nowhere and the ratio $|{cal Z}^perp_f|/|{cal Z}^T_f|$ between the lengths of the normal and tangent components of ${cal Z}$ is constant along $M^m$. It has the emph{principal direction property with respect to ${cal Z}$} if ${cal Z}^T_f$ is an eigenvector of all shape operators of $f$ at all points of $M^m$. In this article we study isometric immersions $fcolon M^mto N^n$ of arbitrary codimension that have either the constant ratio or the principal direction property with respect to distinguished vector fields ${cal Z}$ on space forms, product spaces $Sf^ntimes R$ and $Hy^ntimes R$, where $Sf^n$ and $Hy^n$ are the $n$-dimensional sphere and hyperbolic space, respectively, and, more generally, on warped products $Itimes_{rho}Q_e^n$ of an open interval $Isubset R$ and a space form $Q_e^n$. Starting from the observation that these properties are invariant under conformal changes of the ambient metric, we provide new characterization and classification results of isometric immersions that satisfy either of those properties, or both of them simultaneously, for several relevant instances of ${cal Z}$ as well as simpler descriptions and proofs of some known ones for particular cases of ${cal Z}$ previously considered by many authors.
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