Geometry induced on some minimal surfaces
What we do?
For minimal surfaces: Catenoid and Helicoid 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.
Solution:
| Domain[manifold] | manifold - string - a manifold name or a name of a manifold domain. |
| Metric[id→expr] | id - variable - metric identifier,
expr - expression - metric declaration. |
| Connection[id] | id - variable - connection identifier. |
Necessary functions.
We redefine `atlas/simp` procedure just for right simplification (this is not necessary but it leads to more compact results):
Minimal surfaces
After that we declare constant a:
| Out[9]= |  |
Domain R3
This domain is just 3-dimensional Euclidean space:
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Declare 1-forms for to use them as a coframe:
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Declare vector fields to use them as a frame:
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| Out[14]= |  |
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| Out[16]= |  |
Catenoid
| Out[17]= |  |
Declare 1-forms for coframe:
| Out[18]= |  |
Declare vector fields for frame:
| Out[19]= |  |
Coframe declaration for the catenoid:
| Out[20]= |  |
Frame declaration for the catenoid:
| Out[21]= |  |
Now we declare embedding of the catenoid into the Euclidean space
R3:
| Out[22]= |  |
| Out[23]= |  |
After that we can calculate metric induced on the catenoid by the embedding:
| Out[24]= |  |
Calculation of the corresponding connection and curvature:
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| Out[26]= |  |
Calculation of riemannian and ricci tensors of the embedded catenoid:
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| Out[28]= |  |
Calculation of ricci scalar of the embedded catenoid:
| Out[29]= |  |
We can also calculate the invariants (the second fundamental form and mean curvature vector) of the embedding:
| Out[30]= |  |
Thus the embedding is a minimal one (mean curvature vector is equal to zero):
Let us extract the second fundamental form:
| Out[31]= |  |
Out[32]//MatrixForm= |
| |  |
Now we can calculate the corresponding tensor:
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Helicoid
Now current manifold is a helicoid.
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Declare 1-forms for helicoid coframe:
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Declare vector fields for helicoid frame:
| Out[36]= |  |
Coframe declaration for the helicoid:
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Frame declaration for the helicoid:
| Out[38]= |  |
Now we declare embedding of the helicoid into
R3:
| Out[39]= |  |
| Out[40]= |  |
After that we can calculate metric induced on the helicoid by the embedding:
| Out[41]= |  |
Calculation of the corresponding connection and curvature:
| Out[42]= |  |
| Out[43]= |  |
Calculation of riemannian and ricci tensors of the embedded helicoid:
| Out[44]= |  |
| Out[45]= |  |
Calculation of ricci scalar of the embedded helicoid:
| Out[46]= |  |
Let us calculate the invariants (the second fundamental form and mean curvature vector) of the embedding:
| Out[47]= |  |
Thus the embedding is a minimal one (mean curvature vector is equal to zero):
Let us extract the second fundamental form:
| Out[48]= |  |
Out[49]//MatrixForm= |
| |  |
Now we can calculate the corresponding tensor:
| Out[50]= |  |
Sherk surface
Now current manifold Sherk surface.
| Out[51]= |  |
| Out[52]= |  |
| Out[53]= |  |
Coframe declaration for the surface:
| Out[54]= |  |
Frame declaration for the surface:
| Out[55]= |  |
Now we declare embedding of the surface into
R3:
| Out[56]= |  |
| Out[57]= |  |
After that we can calculate metric induced on the surface by the embedding:
| Out[58]= |  |
Calculation of the corresponding connection and curvature:
| Out[59]= |  |
Returning on a domain
| Out[60]= |  |
Thus we are on the Sherk surface.
We can return on any previous domain easily. Let us return on the catenoid:
| Out[61]= |  |
Suppose we wish to calculate lie derivative of the corresponding metric -
LUj(Gk):
| Out[62]= |  |
Thus
is Killing vector field on the catenoid.
Let us jump on the helicoid and do the same:
| Out[63]= |  |
| Out[64]= |  |
Obviously that
is Killing vector field on the helicoid.
Where are we?
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- Lie derivative
- Exterior product
- Tensor product
- Hodge operator
- Covariant derivative
