Orientable surfaces are two-dimensional manifolds that have a consistent choice of 'direction' at every point, allowing for a continuous distinction between two sides. This property means that if you traverse the surface, you can return to your starting point without flipping over to the opposite side. Understanding orientability is crucial in various geometric contexts, including the application of the Gauss-Bonnet theorem and the analysis of the Euler characteristic, as these concepts often depend on whether a surface can be oriented consistently.
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