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Define manifold as a whole

This notebook illustrates how to define a manifold as a whole using Atlas package. To do this we use 3-sphere as an example.

What we do?

    • 3-dimensional sphere is defined as an Atlas {N, S}=S3 of two charts - north (N) and south (S). Each chart has its own coframe 1-forms, frame vectors etc. One can use the mapping procedure to define chart changes and the pullback operator to transfer forms and tensors from one chart into another.
    Constants[c1,c2,...,ci,...,cn] c1,c2,...,ci,...,cn - constants identifiers.
    Curvature[id] id - variable - curvature identifier.

    Necessary functions.

Sphere:

First of all we load Atlas package:
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Constants declaration:
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Forms declaration:
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Vectors declaration:
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Sphere dimension:
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North chart of the sphere - N.

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Coframe for the north chart:
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Frame for the north chart:
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Metric for the north chart:
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Connection calculation for the north chart:
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Curvature calculation for the north chart:
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Riemanninan tensor calculation:
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Ricci tensor calculation:
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Ricci scalar calculation:
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South chart of the sphere - S.

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Coframe declaration for the south chart:
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Frame declaration for the south chart:
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Metric declaration for the south chart:
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Connection calculation for the south chart:
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Curvature calculation for the south chart:
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Riemannian tensor calculation for the south chart:
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Ricci tensor calculation for the south chart:
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Ricci scalar calculation for the south chart:
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Chart changings - and

Chart changing from S to N: .
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Visualize the mapping
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The mapping info
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Chart changing from N to S: .
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Visualize the mapping
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The mapping info
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Verify calculation of Riemannian and Ricci tensor using restriction operator `&/`:
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For metric tensors we have:
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Let us see who is who:
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