Geometry induced on some minimal surfaces 

Problem: 

For minimal surfaces 

       

calculate the following: first fundamental form (metric tensor field), connection 1-forms, curvature 2-forms, Riemann tensor field, Ricci tensor field, mean curvature vectors, second fundamental form. 

Minimal surfaces 

>

restart:

 

First of all we load atlas package:
with(atlas): 

We redefine `atlas/simp` procedure just for right simplification (this is not necessary but it leads to more compact results):
`atlas/simp`:=proc(a);
collect(simplify(a),`&.`,factor)
end: 

After that we declare constant : 

>	Constants(a);

 

(2.1)

 

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)

 

Connection calculation:
Connection(Gamma); 

(2.1.7)

 

Catenoid 

Now we on a catenoid.
Domain(K); 

(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 catenoid:
Coframe(u[1]=d(zeta),u[2]=d(phi)); 

(2.2.4)

 

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

(2.2.5)

 

Now we declare embedding of the catenoid into the Euclidean space :
Mapping(f,K,R^3,
       x=a*cosh(zeta/a)*cos(phi),
       y=a*cosh(zeta/a)*sin(phi),
       z=zeta); 

 

	(2.2.6)

 

After that we can calculate metric induced on the catenoid by the embedding:
Metric(G[K] = g &/ f); 

(2.2.7)

 

Calculation of the corresponding connection and curvature: 

>	Connection(omega[K]);

 

(2.2.8)

 

>	Curvature(Omega[K]);

 

(2.2.9)

 

Calculation of riemannian and ricci tensors of the embedded catenoid: 

>	Riemann(R[K]);

 

(2.2.10)

 

>	Ricci(ric[K]);

 

(2.2.11)

 

Calculation of ricci scalar of the embedded catenoid:
RicciScalar(s[K]); 

(2.2.12)

 

We can also calculate the invariants (the second fundamental form and mean curvature vector) of the embedding: 

>	Inv[K]:=Invariants(f);

 

(2.2.13)

 

Thus the embedding is a minimal one (mean curvature vector is equal to zero): 

Let us extract the second fundamental form: 

>	B[K]:=eval(Inv[K][secondForm]);

 

(2.2.14)

 

Now we can calculate the corresponding tensor: 

>	'B[K]'=add(add(`&.`(e[i],e[j],eval(B[K])[i,j]) ,i=1..2),j=1..2);

 

(2.2.15)

 

Helicoid 

Now current manifold is a helicoid.
Domain(H); 

(2.3.1)

 

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

(2.3.2)

 

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

(2.3.3)

 

Coframe declaration for the helicoid:
Coframe(v[1]=d(theta),v[2]=d(tau)); 

(2.3.4)

 

Frame declaration for the helicoid:
Frame(V[k]); 

(2.3.5)

 

Now we declare embedding of the helicoid into :
Mapping(psi,H,R^3,
       x=tau*cos(theta),
       y=tau*sin(theta),
       z=a*theta); 

 

	(2.3.6)

 

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

(2.3.7)

 

Calculation of the corresponding connection and curvature: 

>	Connection(omega[H]);

 

(2.3.8)

 

>	Curvature(Omega[H]);

 

(2.3.9)

 

Calculation of riemannian and ricci tensors of the embedded helicoid: 

>	Riemann(R[H]);

 

(2.3.10)

 

>	Ricci(ric[H]);

 

(2.3.11)

 

Calculation of ricci scalar of the embedded helicoid:
RicciScalar(s[H]); 

(2.3.12)

 

Let us calculate the invariants (the second fundamental form and mean curvature vector) of the embedding: 

>	Inv[H]:=Invariants(psi);

 

(2.3.13)

 

Thus the embedding is a minimal one (mean curvature vector is equal to zero): 

Let us extract the second fundamental form: 

>	B[H]:=eval(Inv[H][secondForm]);

 

(2.3.14)

 

Now we can calculate the corresponding tensor: 

>	'B[H]'=add(add(`&.`(e[i],e[j],eval(B[H])[i,j]) ,i=1..2),j=1..2);

 

(2.3.15)

 

 

Sherk surface 

Now current manifold Sherk surface.
Domain(S); 

(2.4.1)

 

Declare 1-forms:
Forms(xi[k]=1); 

(2.4.2)

 

Declare vector fields:
Vectors(Xi[j]); 

(2.4.3)

 

Coframe declaration for the surface:
Coframe(xi[1]=d(alpha),xi[2]=d(beta)); 

(2.4.4)

 

Frame declaration for the surface:
Frame(Xi[k]); 

(2.4.5)

 

Now we declare embedding of the surface into :
Mapping(h,S,R^3,
       x=alpha,
       y=beta,
       z=1/a*ln(cos(a*alpha)/cos(a*beta))); 

 

	(2.4.6)

 

After that we can calculate metric induced on the surface by the embedding:
Metric(G[S] = g &/ h); 

(2.4.7)

 

Calculation of the corresponding connection and curvature: 

>	Connection(omega[S]);

 

(2.4.8)

 

>	Curvature(Omega[S]);

 

(2.4.9)

 

Calculation of riemannian and ricci tensors of the embedded surface: 

>	Riemann(R[S]);

 

(2.4.10)

 

>	Ricci(ric[S]);

 

(2.4.11)

 

Calculation of ricci scalar of the embedded surface:
RicciScalar(s[S]); 

(2.4.12)

 

Let us calculate the invariants (the second fundamental form and mean curvature vector) of the embedding: 

>	Inv[S]:=Invariants(h);

 

(2.4.13)

 

Thus the embedding is a minimal one (mean curvature vector is equal to zero): 

Let us extract the second fundamental form: 

>	B[S]:=eval(Inv[S][secondForm]);

 

(2.4.14)

 

Now we can calculate the corresponding tensor: 

>	'B[S]'=add(add(`&.`(e[i],e[j],eval(B[S])[i,j]) ,i=1..2),j=1..2);

 

(2.4.15)

 

Looking through the results 

Now we can look through the calculated Ricci scalars: 

>

eval(s);

 

(2.5.1)

 

The same for curvature 2-forms: 

>	eval(Omega);

 

(2.5.2)

 

And for the second fundamental forms: 

>

eval(B);

 

(2.5.3)

 

Returning on a domain 

Where are we? 

>

Domain();

 

(2.6.1)

 

Thus we are on the Sherk surface. 

We can return on any previous domain easily. Let us return on the catenoid: 

>	Domain(K);

 

(2.6.2)

 

Suppose we wish to calculate lie derivative of the corresponding metric - : 

>	'L[U[j]](G[K])'=L[U[j]](G[K]);

 

(2.6.3)

 

Thus is Killing vector field on the catenoid. 

Let us jump on the helicoid and do the same: 

>	Domain(H);

 

(2.6.4)

 

>	'L[V[i]](G[H])'=L[V[i]](G[H]);

 

(2.6.5)

 

Obviously that is Killing vector field on the helicoid. 

Where are we? 

>

Domain();

 

(2.6.6)

 

On the helicoid. 

>