Determinantal variety and normal embedding
  The space of matrices of positive determinant GL^+_n inherits an extrinsic metric space structure from R^{n^2}. On the other hand, taking the infimum of the lengths of all paths connecting two points in GL^+_n gives an intrinsic metric. We prove bilipschitz equivalence for intrinsic and extrinsic metrics on GL^+_n, exploiting the conical structure of the stratification of the space of n by n matrices by rank. 
Karin U. Katz, Mikhail G. Katz, Dmitry Kerner, Yevgeny Liokumovich
math.AG math.DG