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Let X be a smooth n-dimensional projective variety embedded in some projective space ℙ N  over the field ℂ of the complex numbers. Associated with the general projection of X to a space ℙ N-m  (N-m>n+1) one defines an extended Gauss map  (in case N-m>2n-1 this is the Gauss map of the image of X under the projection). We prove that  is smooth. In case any two different points of X do have disjoint tangent spaces then we prove that  is injective.

The injectivity of the extended Gauss map of general projections of smooth projective varieties