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In this article, we study constant mean curvature isometric immersions into$\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ and weclassify these isometric immersions when the surface has constant intrinsiccurvature. As applications, we use the sister surface correspondence toclassify the constant mean curvature surfaces with constant intrinsic curvaturein the $3-$dimensional homogenous manifolds $\mathbb{E}(\kappa, \tau)$ and weuse the Torralbo-Urbano correspondence to classify the parallel mean curvaturesurfaces in $\mathbb{S}^2 \times \mathbb{S}^2$ and $\mathbb{H}^2 \times\mathbb{H}^2$ with constant intrinsic curvature. It is worthwhile to point outthat these classifications provide new examples.

Constant mean curvature Isometric Immersions into $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ and related results