Projective spaces on normed division algebras

In this work firstly the real projective space RPn−1 is defined as a quotient space of R^n − {0}. This space also defined as a quotient space of the sphere Sn−1 via the map P. It is shown that this space is (n − 1) dimensional differentiable manifold and it is proven that RP1 ∼= S1 in case of n = 2 . Some other interpretations are given for RP^2 . Secondly the complex projective space CP^(n−1) is defined as a quotient space of C^n − {0} . This space also considered as a quotient space of the sphere S^(2n−1) via the map P . It is shown that CP^(n−1) is (2n−2) dimensional differentiable manifold and it is proven that CP1 ∼= S2 in case of n = 2 . Then the quaternionic projective space HP^(n−1) is defined as a quotient space of H^n −{0} . The quaternionic projective space also defined as the quotient space of S^(4n−1) via the map P. It is shown that HP^(n−1) is (4n−4) dimensional differentiable manifold and it is proven that HP1 ∼= S^4 in case of n = 2 . Lastly the octonionic projective spaces OP^1 and OP^2 are defined by equivalence relations which are completely different from the previously defined . It is proven that these spaces are differentiable manifolds of dimension 8 and 16 respectively .