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We consider constant mean curvature surfaces (invariant by a continuous groupof isometries) lying at bounded distance from a horizontal geodesic on anyhomogeneous $3$-manifold $\mathbb{E}(\kappa,\tau)$ with isometry group ofdimension $4$. These surfaces are called horizontal tubes. We show that theyfoliate $\mathbb{E}(\kappa,\tau)$ minus one or two horizontal geodesicsprovided that $(1-x_0^2)\kappa+4\tau^2\leq 0$, where $x_0\approx 0.833557$. Wealso describe precisely how horizontal and vertical geodesics get deformed byDaniel's sister correspondence and conclude that the family of horizontal tubesis preserved by the correspondence. These tubes are topologically tori in$\mathbb{S}^2\times\mathbb{R}$ and Berger spheres, in which case we computetheir conformal type and analyze numerically their isoperimetric profiles.

Invariant constant mean curvature tubes around a horizontal geodesic in $\mathbb{E}(\kappa,\tau)$-spaces