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Atlas 2 Logo

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This notebook illustrates how to use the Atlas package to solve problems in elementary differential geometry. As an example we find the geometry of the Atlas 2 Logo surface.

What we do?

  • We assume that the geometry is induced by the corresponding embedding of a surface into the flat Euclidian 3-dimensional space.
    • We calculate the connection, curvature, curvature tensor field (Riemann tensor field), Ricci tensor field and Ricci scalar function which is proportional to scalar curvature.
    Load atlas package:
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    Atlas 2 Logo surface:
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Solution:

Forms[f1→n,f2→k,...,fi→p]fi→p -rules where fi - form identifier and p is a variable or an integer - the form's degree.
Vectors[v1,v2,...,vi,...,vn]vi - vector identivier.
Coframe[id1→ expr1,id2→ expr2,...idn→ exprn]id - identifier for indexed variable - the coframe 1-forms. n - dimension of working manifold (a variable or integer). idi→ expri - equation where idi is indexed variable - coframe 1-form and expri is decomposition of the 1-form on exact 1-forms.
Metric[id→expr]id - variable - metric identifier. expr - expression - metric declaration.
Connection[id]id - variable - connection identifier.

Necessary functions.

Description of the total space R3

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 is equal to zero of course).
Define Euclidean space as a manifold:
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Declare 1-forms for the space coframe:
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Declare vectors for the space frame:
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Declare coframe on the space:
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Declare frame on the space:
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Declare flat metric on the space:
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Calculate connection of the metric:
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Curvature[id]id - variable - curvature identifier.
Riemann[id]id - variable - corresponding identifier.
Ricci[id]id - variable - corresponding identifier.
RicciScalar[id]id - variable - corresponding identifier.

Necessary functions.

Atlas 2 Logo

Define the Atlas 2 Logo as a manifold:
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Declare constants a, b, c, d:
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Declare functions:
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Declare 1-forms for logo coframe:
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Declare vectors for logo frame:
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Declare coframe on the logo:
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Declare frame of the surface:
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Declare mapping of the Atlas 2 Logo into R2:
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Calculate metric on the logo using Pullback- operator:
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Calculate connection of the embedding:
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Calculate curvature of the embedding:
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Calculate Riemann tensor field:
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Calculate Ricci tensor field:
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Calculate Ricci scalar curvature:
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