Principal Bundle Structure of Matrix Manifolds

In this paper, we introduce a new geometric description of the manifolds matrices fixed rank. The starting point is Grassmann manifold Gr(Rk) linear subspaces dimension r<k in Rk, which avoids use equivalence classes. set equipped with an atlas, provides it structure analytic modeled on R(k?r)×r. Then, define atlas for Mr(Rk×r) full rank and prove that resulting principal bundle base typical fibre GLr, general group invertible Rk×k. Finally, Mr(Rn×m) non-full Gr(Rn)×Gr(Rm) GLr. indexed itself, allows natural definition neighbourhood given matrix, being proved to possess Lie group. Moreover, topology induced by proven be embedded submanifold matrix space Rn×m subspace topology. proposed then results Rn×m, seen as union Mr(Rn×m), continuous map.