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We consider surfaces with parallel mean curvature vector field and finitetotal curvature in product spaces of type $\mathbb{M}^n(c)\times\mathbb{R}$,where $\mathbb{M}^n(c)$ is a space form, and characterize certain of thesesurfaces. When $n=2$, our results are similar to those obtained in \cite{bds}for surfaces with constant mean curvature in space forms.

Constant Mean Curvature Surfaces in $\mathbb{M}^2(c)\times\mathbb{R}$ and Finite Total Curvature