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Invariants - calculation of a mapping invariants

Invariants[f]
allows one to calculate invariants of a mapping between manifolds (see Mapping)
  • f - mapping identifier.
  • If mapping f is the embedding of a curve then the curve's normalized moving frame and the curve's curvatures are calculated.
  • If mapping f is an embedding or immersion then the second fundamental form and mean curvature vector are calculated.
  • If mapping f is a submersion then the A and T invariants are calculated. In that case, some additional calculations are performed: the mean curvature vector of corresponding fibers, the integrability obstruction of corresponding horizontal distribution and the riemannian obstruction (if the submersion is not a riemannian one).
  • The corresponding rules are as follows:
  • Let mapping F:MN be declared by functions (see Mapping):
  • where m=dim(M), n=dim(N); {x1, x2, ..xm} are local coordinates on M and {y1, y2, ..yn} are local coordinates on N.
Curve
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:
where p=0..n-1 and
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.