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Orientability of Regular Zero Set.
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A manifold M of dim n is orientable if there exists a smooth nowhere vanishing n form on M. On a regular zero set in R^{n+1}, there is always a nowhere vanishing n-form so Regular Zero sets in R^{n+ 1} are orientable manifolds. For example if S is cut out by f in R^3 then differentiating (f dy = 0) gives f_x dx dy = f_z dy dz. Thus we have dx dy/ f_z = dy dz/ f_x = dx dz / f_y whenever this expression makes sense. Gluing them gives a nowhere vanishing 2 form on S. In particular the sphere S2 is an orientable Manifold. However, it does not have continuous global frame since every continuous vector field on an even dimensional sphere must vanish somewhere.
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