atlas[Projectors] - calculation of a mapping projectors (tangent and normal or horizontal and vertical)

Calling Sequence:

     Projectors(F)

Parameters:

      F - mapping identifier

Description:

The Projectors  procedure calculates projectors of a mapping between manifolds (see atlas[Mapping] ). If mapping F is an embedding  or immersion  then normal  and tangent  projectors are calculated. If mapping F is a submersion  then horizontal  and vertical  projectors are calculated. 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.
• If  then the mapping is treated as an embedding or immersion thus normal and tangent projectors are calculated.
• If  then the mapping is treated as a submersion thus horizontal and vertical projectors are calculated.
• The procedure returns a table with corresponding indexes ( normal, tangential, horizontal, vertical ) and entries. The entries are corresponding tensors of type [1,1] (see examples below).
• It should be pointed out that the calculation is only available if actual metric are declared in both manifolds. To get more information see examples : Ricci-flat warped product  

Examples:
restart:
with(atlas):

Just for right simplification:
`atlas/simp`:=proc(a)
subs({cos(phi)^2=1-sin(phi)^2},a);
normal(%);
end:

Domain

This domain is just 3-dimensional Euclidean space:
Domain(R^3);

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 flat metric:
: Metric(g=d(x)&.d(x)+d(y)&.d(y)+d(z)&.d(z));

Domain

This domain is 1-sphere (a circle).
Domain(S);

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

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

Coframe declaration for the sphere:
Coframe(u[1]=d(phi));

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

Mappings

Mapping of the sphere into the Euclidean space :
Mapping(psi,S,R^3,
        x=cos(phi),
        y=sin(phi),
        z=0);

Mapping of the Euclidean space / Z (Z axis is excluded) into the sphere:
Mapping(pi,R^3,S,
        phi=arctan(y/x));

Metric on the sphere

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

  projectors

Calculation of the projection:
P[psi]:=Projectors(psi);

Normal projector extraction
N[psi]:=P[psi][normal];

Tangent projection extraction:
T[psi]:=P[psi][tangent];

Jump into the previous domain:
Domain(R^3);

Verify that there are no tangent or normal vectors among frame ones:
'iota[E[i]](T[psi])'=iota[E[i]](T[psi]);

'iota[E[i]](N[psi])'=iota[E[i]](N[psi]);

Verify that "rotation" vector  is tangent one:
Z:=cos(phi)*E[2]-sin(phi)*E[1];
'iota[Z](T[psi])'=simplify(iota[Z](T[psi]));
'iota[Z](N[psi])'=simplify(iota[Z](N[psi]));

  projectors

Calculation of the projectors:
P[pi]:=Projectors(pi);

Vertical projector extraction:
V[pi]:=P[pi][vertical];

Horizontal projector extraction:
H[pi]:=P[pi][horizontal];

Jump into :
Domain(R^3);

Verify that there are no tangent or normal vectors among frame ones:
'iota[E[i]](V[pi])'=iota[E[i]](V[pi]);

'iota[E[i]](H[pi])'=iota[E[i]](H[pi]);

Verify that "rotation" vector  is horizontal one:
X:=x*E[2]-y*E[1];
'iota[X](V[pi])'=simplify(iota[X](V[pi]));
'iota[X](H[pi])'=simplify(iota[X](H[pi]));

See Also:

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