Einstein submanifolds with parallel mean curvature into $\mathbb{S}^n\times\mathbb{R}

We prove that Einstein submanifolds in $\mathbb{S}^n\times\mathbb{R}$ with flat normal bundle and parallel mean curvature are warped product of isometric immersions. Key words: Einstein submanifolds, Parallel mean curvature, Flat normal bundle.