Minimality of invariant submanifolds in Metric Contact Pair Geometry

TL;DR

This paper investigates conditions under which invariant submanifolds in metric contact pair manifolds are minimal, providing new criteria based on invariance, transversality, and complex structure integrability.

Contribution

It establishes new minimality conditions for invariant submanifolds in metric contact pair geometry, including cases with normal and complex structures, extending previous understanding.

Findings

Invariant submanifold tangent to a Reeb vector field and orthogonal to the other is minimal.

A transverse invariant submanifold is minimal if the angle with Reeb vector fields is constant.

Complex submanifolds tangent to both Reeb vector fields are minimal.

Abstract

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.