atlas for Maple package: Mappings and its Invariants

atlas[Invariants] - calculation of a mapping invariants

Description:

The Invariants procedure allows one to calculate invariants of a mapping between manifolds (see atlas[Mapping]). 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: 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 the actual metric and connection are declared on an N manifold.

Embedding

If then the mapping is treated as an embedding or immersion, thus the 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 the actual metric and the connection are declared (calculated) on the N manifold and

that the actual metric is declared (calculated) on the M manifold.

Submersion

If then the mapping is treated as a submersion, thus the 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 the actual metric and connection are declared on the 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);

(2.1.1)

Declare constant :
Constants(Lambda);

${`+`(`-`(I)), I, Pi, _Z, Catalan, Lambda}$

(2.1.2)

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

(2.1.3)

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

(2.1.4)

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

(2.1.5)

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

(2.1.6)

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);

(2.1.7)

Calculate connection:
Connection(omega);

(2.1.8)

Domain

>	Domain(S^2);

(2.2.1)

Declare constant :
Constants(lambda);

${`+`(`-`(I)), I, Pi, _Z, Catalan, Lambda, lambda}$

(2.2.2)

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

(2.2.3)

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

(2.2.4)

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

(2.2.5)

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

(2.2.6)

Domain

>	Domain(S);

(2.3.1)

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

(2.3.2)

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

(2.3.3)

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

(2.3.4)

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

(2.3.5)

Mappings

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


	(2.4.1)

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


	(2.4.2)

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));