Surface geometry  

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

Description:

This worksheet illustrates how to use atlas  package to solve problems in elementary differential geometry. As an example we find geometry  of the following surfaces:

We assume that the geometry is induced by the corresponding embedding of a surface into the flat Euclidian 3-dimensional space. Thus we construct corresponding embedding and calculate the following quantities: first fundamental form  ( metric  tensor field), second fundamental form and field of mean curvature vectors . Besides that we calculate connection 1-forms , curvature 2-forms , curvature tensor field  ( Riemann   tensor  field), Ricci tensor  field and Ricci scalar  function which is proportional to scalar curvature .

Solution:

Load atlas package:

restart:
with(atlas):

Just for right simplification:
`atlas/simp`:=proc(a) factor(simplify(a,trig)) end:
 

Description of the total space

First of all we have to describe the space we are working in. The space is 3-dimensional Euclidean (flat) space. To define the space we declare domain , forms , vectors , coframe , frame , flat metric  and calculate connection  (it equals to zero of cause).

Define Euclidean space  as a manifold :
Domain(R^3);

Declare 1-forms  for the space coframe:
Forms(e[k]=1);

Declare vectors  for the space frame:
Vectors(E[j]);

Declare coframe  on the space:
Coframe(e[1]=d(x),e[2]=d(y),e[3]=d(z));

Declare frame  on the space:
Frame(E[k]);

Declare flat metric  on the space:
Metric(g[R]=d(x)&.d(x)+d(y)&.d(y)+d(z)&.d(z));

Calculate connection  of the metric:
Connection(Gamma);

  Torus

Define the torus as a manifold :
Domain(T);

Declare constants  a, b:
Constants(a,b);

Declare 1-forms  for torus coframe
Forms(w[i]=1);

Declare vectors  for torus frame:
Vectors(W[k]);

Declare coframe  on the torus:
Coframe(w[1]=d(alpha),w[2]=d(beta));

Declare frame  of the surface:
Frame(W[l]);

Declare mapping  of the torus into :
Mapping(pi,T,R^3,x=(a+b*cos(beta))*cos(alpha),
                 y=(a+b*cos(beta))*sin(alpha),
                 z=b*sin(beta));

Calculate metric  on the torus using `&/`  operator:
Metric(g[T] = g[R] &/ pi);

Calculate invariants  of the mapping:
Inv[T]:=Invariants(pi);

Extract second fundamental form  as a table:
sF:=eval(Inv[T][secondForm]);

Construct second fundamental form  as corresponding tensor field ( `&.`  -  is tensor product operator):
B:=add(add(`&.`(e[i],e[j],sF[i,j]),i=1..3),j=1..3);

Extract mean curvature vector :
mu:=Inv[T][meanCurvature];

Calculate mean curvature :
chi:=simplify(g[R](mu,mu));

Use differential operator  `&D` to express tangent vectors on the torus  by total space frame vectors :
W1:=W[1] &D pi;
W2:=W[2] &D pi;

Verify that mean curvature vector field is normal to the tangent space  of the torus:
'g[R](mu,W1)'=simplify(g[R](mu,W1));
'g[R](mu,W2)'=simplify(g[R](mu,W2));

Calculate connection  of the embedding:
Connection(omega[T]);

Calculate curvature  of the embedding:
Curvature(Omega[T]);

Calculate Riemann  tensor field:
Riemann(R[T]);

Calculate Ricci  tensor field:
Ricci(ric[T]);

Calculate Ricci scalar  curvature:
RicciScalar(scal[T]);

  Helicoid

Define the surface as a manifold:
Domain(H);

Declare 1-form for surface coframe
Forms(s[i]=1);

Declare vectors for surface frame:
Vectors(S[k]);

Declare coframe on the surface:
Coframe(s[1]=d(u[h]),s[2]=d(v[h]));

Declare frame of the surface:
Frame(S[k]);

Declare mapping of the surface into :
Mapping(psi,H,R^3,x=u[h]*cos(v[h]),
                  y=u[h]*sin(v[h]),
                  z=b*v[h]);

One can also calculate metric induced on the surface by the mapping.

Metric(g[H] = g[R] &/ psi);

Calculate invariants of the mapping:
Inv[H]:=Invariants(psi);

It is well known that helicoid is a minimal surface . Thus nothing surprising about zero mean curvature vector field .

Calculate connection of the embedding:
Connection(omega[H]);

Calculate curvature of the embedding:
Curvature(Omega[H]);

Calculate Riemann tensor field:
Riemann(R[H]);

Calculate Ricci tensor field:
Ricci(ric[H]);

Calculate Ricci scalar curvature:
RicciScalar(scal[H]);

  Abstract revolution surface

Define the surface as a manifold:
Domain(A);

Declare functions:
Functions(rho=rho(u),zeta=zeta(u));

Declare 1-form for surface coframe
Forms(p[i]=1);

Declare vectors for surface frame:
Vectors(P[k]);

Declare coframe on the surface:
Coframe(p[1]=d(u),p[2]=d(v));

Declare frame of the surface:
Frame(P[l]);

Declare mapping of the surface into :
Mapping(f,A,R^3,x=rho*cos(v),
                y=rho*sin(v),
                z=zeta);

One can also calculate metric induced on the surface by the mapping.

Metric(g[A] = g[R] &/ f);

Calculate invariants of the mapping:
Inv[A]:=Invariants(f);

Calculate connection of the embedding:
Connection(omega[A]);

Calculate curvature of the embedding:
Curvature(Omega[A]);

Calculate Riemann tensor field:
Riemann(R[A]);

Calculate Ricci tensor field:
Ricci(ric[A]);

Calculate Ricci scalar curvature:
RicciScalar(scal[A]);