atlas[Invariants]  - calculation of a mapping invariants

Calling Sequence:

     Invariants(F)

Parameters:

      F - mapping identifier

Description:

The Invariants  procedure calculates invariants of a mapping between manifolds (see atlas[Mapping] ). If mapping F is 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 second fundamental form   and mean curvature vector  are calculated. If mapping F is a submersion  then A  and T  invariants are calculated. In that case some additional calculations are performed: mean curvature vector  of corresponding fibers, intagrability obstruction  of corresponding horizontal distribution and riemannian obstruction  (if the submersion is not a riemannian one). The corresponding rules are as follows:

• Let mapping F:   be declared by functions (see atlas[Mapping] ):
  
  
  
  where   ;    are local coordinates on M and    are local coordinates on N.

Curve

If  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  and

     are basis  vectors of   normalized moving frame of the embedded curve;

     are curvatures  of the embedded curve (  ).

If N manifold is 3-dimensional Euclidean space than  is the curvature of the curve and  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 actual metric and connection are declared on N manifold.

Embedding

If  then the mapping is treated as an embedding or immersion thus second fundamental form and mean curvature vector are calculated. If N and T are corresponding normal and tangential projectors (see atlas[Projectors] ) then the embedding (or immersion) invariants are defined as follows.
For any vector fields X and Y on M we have:
       for second fundamental form :
       for mean curvature vector : .

 It should be pointed out that the calculation is only available if actual metric and connection are declared (calculated) on N manifold and actual metric is declared (calculated) on M manifold.

Submersion

If  then the mapping is treated as a submersion thus mean curvature vectors, A and T invariants, riemannian and integrability obstructions are calculated. If  and  are corresponding horizontal and vertical projectors then the submersion invariants are defined as follows.
For any vector fields X and Y on M we have:
      for tensor A : ;
      for tensor T : ;
      for meanCurvature vector : ;
      for integrabilityObstruction : ;
      for   riemannianObstruction : .

 It should be pointed out that the calculation is only available if actual metric and connection are declared on M manifold.

• The procedure returns a table with corresponding indexes (

  meanCurvature, A, T, secondForm, riemannianObstruction, integrabilityObstruction, curvatures, basis ) and entries. The entries are corresponding values (see examples below).

• To get more information see examples .

Examples:
restart:
with(atlas):

Domain :
Domain(S^3);

Declare constant :
Constants(Lambda);

Declare 1-forms for to use them as a coframe:
Forms(e[j]=1);

Declare vector fields to use them as a frame:
Vectors(E[i]);

Declare coframe 1-forms:
Coframe(e[1]=d(x),e[2]=d(y),e[3]=d(z));

Declare frame vectors:
Frame(E[j]);

Declare metric:
Metric(g=4*(d(x)&.d(x)+d(y)&.d(y)+d(z)&.d(z))/(1+Lambda*(x^2+y^2+z^2))^2);

Calculate connection:
Connection(omega);

Domain
Domain(S^2);

Declare constant :
Constants(lambda);

Declare 1-forms for coframe:
Forms(u[k]=1);

Declare vector fields for frame:
Vectors(U[j]);

Coframe declaration:
Coframe(u[1]=d(zeta),u[2]=d(xi));

Frame declaration for the sphere:
Frame(U[k]);

Domain
Domain(S);

Declare 1-forms for coframe:
Forms(w[k]=1);

Declare vector fields for frame:
Vectors(W[j]);

Coframe declaration:
Coframe(w[1]=d(tau));

Frame declaration for the sphere:
Frame(W[k]);

Mappings

Simple mapping of the 2-sphere  into 3-sphere  (embedding):
Mapping(psi,S^2,S^3,
        x=zeta,
        y=xi,
        z=lambda);

Simple mapping of the 3-sphere  into 2-sphere    (submersion):
Mapping(pi,S^3,S^2,
        zeta=x^2+y^2,
        xi=z);

Simple mapping of the 1-sphere  into 3-sphere  (curve):
Mapping(phi,S,S^3,
        x=1/sqrt(2*Lambda)*cos(tau),
        y=1/sqrt(2*Lambda)*sin(tau),
        z=1/sqrt(2*Lambda));

Metric and connection induced on 2-sphere -  by  embedding

Jump to 2-sphere:
Domain(S^2);

After that we can calculate metric induced on  2-sphere  by  embedding:
Metric(G = g &/ psi);

Connection(Gamma);

   - invariants (embedding)

Calculation of the invariants:
Inv[psi]:=Invariants(psi);

   - invariants (submersion)

Calculation of the invariants:
Inv[pi]:=Invariants(pi);

   - invariants (curve)

To simplify the results:
`atlas/simp`:=proc(a) factor(simplify(a)) end:

Calculation of the invariants:
Inv[phi]:=Invariants(phi);

See Also:

atlas , atlas[Domain] , atlas[`&/`] , atlas[Projectors] .