Define manifold as a whole 

Description: 

This worksheet illustrates how to define a manifold as a whole using atlas package. To do this we use 3-sphere as an example. 

3-dimensional sphere is defined as an atlas of two charts - north (N)  and south (S). Each chart has its own coframe 1-forms, frame vectors etc. 

One can use the mapping procedure to define chart changes and the pullback operator to transfer forms and tensors from one chart into another.   

Sphere: 

>	restart: with(atlas):

 

Constants declaration:
Constants(lambda); 

(2.1)

 

Forms declaration:
Forms(e[j]=1,u[k]=1); 

(2.2)

 

Vectors declaration:
Vectors(E[i],U[j],X,Y,Z); 

(2.3)

 

Sphere dimension:
n:=3; 

(2.4)

 

North chart of the sphere - N. 

>	Domain(N);

 

(2.1.1)

 

Coframe for the north chart:
Coframe(e[i]=d(x[i]),i=1..n); 

(2.1.2)

 

Frame for the north chart:
Frame(E[j]); 

(2.1.3)

 

Metric for the north chart:
Metric(g[N]=add(4*d(x[i])&.d(x[i])/(1+lambda*add(x[j]^2,j=1..n))^2,i=1..n)); 

(2.1.4)

 

Connection calculation for the north chart:
Connection(omega[N]); 

(2.1.5)

 

Curvature calculation for the north chart:
Curvature(Omega[N]); 

(2.1.6)

 

Riemanninan tensor calculation:
Riemann(R[N]); 

(2.1.7)

 

Ricci tensor calculation:
Ricci(r[N]); 

(2.1.8)

 

Ricci scalar calculation:
RicciScalar(s[N]); 

(2.1.9)

 

South chart of the sphere - S. 

>	Domain(S);

 

(2.2.1)

 

Coframe declaration for the south chart:
Coframe(u[i]=d(y[i]),i=1..n); 

(2.2.2)

 

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

(2.2.3)

 

Metric declaration for the south chart:Metric(g[S]=add(4*d(y[i])&.d(y[i])/(1+lambda*add(y[j]^2,j=1..n))^2,i=1..n)); 

(2.2.4)

 

Connection calculation for the south chart:
Connection(omega[S]); 

(2.2.5)

 

Curvature calculation for the south chart:
Curvature(Omega[S]); 

(2.2.6)

 

Riemannian tensor calculation for the south chart:
Riemann(R[S]); 

(2.2.7)

 

Ricci tensor calculation for the south chart:
Ricci(r[S]); 

(2.2.8)

 

Ricci scalar calculation for the south chart:
RicciScalar(s[S]); 

(2.2.9)

 

Chart changings - and  

Chart changing from S to N: . 

>	Mapping(phi,S,N,seq(x[i]=y[i]/lambda/add(y[j]^2,j=1..n),i=1..n));

 

 

	(2.3.1)

 

Chart changing from N to S: . 

>	Mapping(psi,N,S,seq(y[i]=x[i]/lambda/add(x[j]^2,j=1..n),i=1..n));

 

 

	(2.3.2)

 

Verify calculation of Riemannian and Ricci tensor using restriction operator `&/`: 

>	R[S] - (R[N] &/ phi);

 

(2.3.3)

 

>	r[S] - (r[N] &/ phi);

 

(2.3.4)

 

>	R[N] - (R[S] &/ psi);

 

(2.3.5)

 

>	r[N] - (r[S] &/ psi);

 

(2.3.6)

 

For metric tensors we have: 

>	g[S] = g[N] &/ phi;

 

(2.3.7)

 

>	g[N] = g[S] &/ psi;

 

(2.3.8)

 

Let us see who is who:
Who(); 

(2.5)

 

>

 

See Also:  

atlas[types], atlas[simp], atlas.