Riemann surface
Orientability

Each Riemann surface, being a complex manifold, is orientable as a real manifold. For complex charts f and g with transition function h = f (g (z)), h can be considered as a map from an open set of R to R whose Jacobian in a point z is just the real linear map given by multiplication by the complex number h ' (z). However, the real determinant of multiplication by a complex number α equals | α |, so the Jacobian of h has positive determinant. Consequently, the complex atlas is an oriented atlas.
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