Geometry induced on some minimal surfaces

Copyright © 2004-2008 by  DigiArea Group . All rights reserved.

Problem:

For minimal surfaces: Helicoid, Catenoid and Scherk surface

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);

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));

Connection calculation:
Connection(Gamma);

Catenoid

Now we on a catenoid.
Domain(K);

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

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

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

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

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);

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

Calculation of the corresponding connection and curvature:
Connection(omega[K]);

Curvature(Omega[K]);

Calculation of riemannian and ricci tensors of the embedded catenoid:
Riemann(R[K]);

Ricci(ric[K]);

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

We can also calculate the invariants (the second fundamental form and mean curvature vector) of the embedding:
Inv[K]:=Invariants(f);

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]);

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);

Helicoid

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

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

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

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

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

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

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

Calculation of the corresponding connection and curvature:
Connection(omega[H]);

Curvature(Omega[H]);

Calculation of riemannian and ricci tensors of the embedded helicoid:
Riemann(R[H]);

Ricci(ric[H]);

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

Let us calculate the invariants (the second fundamental form and mean curvature vector) of the embedding:
Inv[H]:=Invariants(psi);

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]);

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);

Sherk surface

Now current manifold Sherk surface.
Domain(S);

Declare 1-forms:
Forms(xi[k]=1);

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

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

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

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)));

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

Calculation of the corresponding connection and curvature:
Connection(omega[S]);

Curvature(Omega[S]);

Calculation of riemannian and ricci tensors of the embedded surface:
Riemann(R[S]);

Ricci(ric[S]);

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

Let us calculate the invariants (the second fundamental form and mean curvature vector) of the embedding:
Inv[S]:=Invariants(h);

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]);

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);

Calculated Ricci scalars:
eval(s);