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.

Load the Atlas package:
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We redefine `atlas/simp` procedure just for right simplification (this is not necessary but it leads to more compact results):
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Minimal surfaces

After that we declare constant a:
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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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Declare coframe 1-forms:
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Declare frame vectors:
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Declare flat metric:
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Connection calculation:
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Catenoid

Now we on a catenoid.
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Declare 1-forms for coframe:
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Declare vector fields for frame:
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Coframe declaration for the catenoid:
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Frame declaration for the catenoid:
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Now we declare embedding of the catenoid into the Euclidean space R3:
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Visualize the mapping:
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After that we can calculate metric induced on the catenoid by the embedding:
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Calculation of the corresponding connection and curvature:
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Calculation of riemannian and ricci tensors of the embedded catenoid:
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Calculation of ricci scalar of the embedded catenoid:
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We can also calculate the invariants (the second fundamental form and mean curvature vector) of the embedding:
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Thus the embedding is a minimal one (mean curvature vector is equal to zero): Let us extract the second fundamental form:
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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:
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Coframe declaration for the helicoid:
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Frame declaration for the helicoid:
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Now we declare embedding of the helicoid into R3:
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Visualize the mapping:
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After that we can calculate metric induced on the helicoid by the embedding:
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Calculation of the corresponding connection and curvature:
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Calculation of riemannian and ricci tensors of the embedded helicoid:
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Calculation of ricci scalar of the embedded helicoid:
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Let us calculate the invariants (the second fundamental form and mean curvature vector) of the embedding:
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Thus the embedding is a minimal one (mean curvature vector is equal to zero): Let us extract the second fundamental form:
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Now we can calculate the corresponding tensor:
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Sherk surface

Now current manifold Sherk surface.
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Declare 1-forms:
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Declare vector fields:
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Coframe declaration for the surface:
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Frame declaration for the surface:
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Now we declare embedding of the surface into R3:
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Visualize the mapping:
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After that we can calculate metric induced on the surface by the embedding:
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Calculation of the corresponding connection and curvature:
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Returning on a domain

Where are we?
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Thus we are on the Sherk surface. We can return on any previous domain easily. Let us return on the catenoid:
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Suppose we wish to calculate lie derivative of the corresponding metric - LUj(Gk):
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Thus is Killing vector field on the catenoid. Let us jump on the helicoid and do the same:
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Obviously that is Killing vector field on the helicoid. Where are we?
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On the helicoid.