Surfaces for which one can find a Unitary normal vector field (remember that a vector field is defined to be differentiable) globally.
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if a surface has non-zero Mean curvature everywhere then it is orientable. To see this imagine that the normal is chosen so that the mean curvature is positive everywhere at every local chart (the other choice would be negative everywhere). However, if it's non-orientable, there is an overlap of charts, where such a choice of normal would be insconsistent in the sense that from one chart it is negative the vector from the other chart. However, by assumption, the mean curvature was positive in one of the charts, and so in the other, as the normal is pointing in the opposite direction, the mean curvature must be negative. This contradicts the assumption that it was positive in every chart. This fact is used in the proof of Jellet-Liebmann theorem