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 allows one to calculate 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 the actual metrics declared in both manifolds. To get more information see examples: Ricci-flat warped product and Geometry induced on some minimal surfaces

 

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); 

(2.1.1)

 

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

(2.1.2)

 

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

(2.1.3)

 

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

(2.1.4)

 

Declare frame vectors:
Frame(E[j]); 

(2.1.5)

 

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

(2.1.6)

 

Domain  

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

(2.2.1)

 

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

(2.2.2)

 

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

(2.2.3)

 

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

(2.2.4)

 

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

(2.2.5)

 

Mappings 

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

 

	(2.3.1)

 

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

 

	(2.3.2)

 

Metric on the sphere 

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

(2.4.1)

 

projectors 

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

(2.5.1)

 

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

(2.5.2)

 

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

(2.5.3)

 

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

(2.5.4)

 

Verify that there are no tangent or normal vectors among frame ones: 

>	'iota[E[i]](T[psi])'=iota[E[i]](T[psi]);

 

(2.5.5)

 

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

 

(2.5.6)

 

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])); 

 

 

	(2.5.7)

 

projectors 

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

(2.6.1)

 

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

(2.6.2)

 

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

(2.6.3)

 

Jump into :
Domain(R^3); 

(2.6.4)

 

Verify that there are no tangent or normal vectors among frame ones: 

>	'iota[E[i]](V[pi])'=iota[E[i]](V[pi]);

 

(2.6.5)

 

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

 

(2.6.6)

 

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])); 

 

 

	(2.6.7)

 

>

 

See Also:  

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