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

We consider surfaces with parallel mean curvature vector field and finite total 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 these surfaces. When (n=2), our results are similar to those obtained in Bérard et al. (Ann Glob Anal Geom 16(3):273–290, 1998) for surfaces with constant mean curvature in space forms.