Invariant submanifolds of metric contact pairs

TL;DR

This paper investigates the geometric properties of invariant submanifolds within metric contact pairs, revealing conditions under which they are minimal, inherit contact structures, and relate to the Reeb vector fields, with implications for their curvature and product structures.

Contribution

It characterizes invariant submanifolds in metric contact pairs, establishing conditions for minimality, contact metric inheritance, and product structures involving Reeb vector fields.

Findings

Invariant submanifolds make constant angles with Reeb vector fields.

Normal case submanifolds of dimension ≥ 2 are minimal.

Leaves of characteristic foliations are minimal.

Abstract

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 $\mathbb{R}$.