I firstly learned about duality in context of differentiable manifolds. Here, we have tangent vectors populating the tangent space and differential forms in its co-tangent counterpart. Acting upon each other a vector and a form produce a scalar (contraction operation). Later, I run into the term of «duality» reading textbook on projective geometry. But here it is used somewhat differently, one talks about duality between points and straight lines in projective plane (point and planes in the case of projective space). Point and lines in ℙ reverse their roles when mapped into ℙ*. I don't quite understand how this two definitions of duality are related. Can they be formulated in similar words and notions (preferably using «language of manifolds»)? How can a point of projective space be equivalent for tangent vector and line for form? What is the equivalent of manifold itself in case of projective geometry?