The Euler characteristic of a compact orientable genus g surface X is nonzero if g is not 1 (the general formula is 2-2g), so the Euler class e(TX) of the tangent bundle TX of X is nonzero. This implies that TX cannot have a continuous nowhere-vanishing section. If it did, e(TX) would be 0. The single-holed torus is the only orientable compact surface that possesses a continuous nonvanishing vector field.