Given a vector space V of dimension n and a natural number k < n, the grassmannian G_k(V) is defined as the set of all subspaces W ⊂ V such that dim(W) = k. In the case of V = Rⁿ, G_k(V) is the set of k-fl ats in Rⁿ and is called real grassmannian [1]. Recently the study of these manifolds has found applicability in several areas of mathematics, especially in Modern Differential Geometry and Algebraic Geometry. This work will build two differential structures on the real grassmannian, one of which is obtained as a quotient space of a Lie group [1], [3], [2], [7]