Alright, if I am getting this right - Say you have a n-dimensional surface in (certainly m>n). Choose a point on the surface and consider the tangent hyperplane at that point. This has n vectors as basis: . To complete this set to a basis of , you need m-n linearly independent vectors, orthogonal to that hyperplane: . These are called principal normal vectors. Divide by their lengths to get them to be unit:2) a principle unit normal vector to the surface.