If
m=1 then the mapping is treated as a curve embedding, thus the curve's normalized moving frame and the corresponding curvatures are calculated. The corresponding invariants satisfy the equations:
_{p} are basis vectors of normalized moving frame of the embedded curve;
_{p} are curvatures of the embedded curve (
_{0}=0, _{n}=0 )
If N manifold is 3-dimensional Euclidean space than
_{1} is the curvature of the curve and
_{2} is the torsion. In that case the sign of the torsion is the same for left or right - handed curves just because in the presented algorithm the moving frame is right-handed for right-handed curve and left-handed for left-handed curve.
It should be pointed out that the calculation is only available if the actual metric and connection are declared on an N manifold.