Conformally flat metric on 2-dimentional sphere

This notebook illustrates how to use Atlas package to solve problems in elementary differential geometry. As an example we discuss conformally flat metric on 2-dimensional sphere.

What we do?

  • We construct the metric on sphere and calculate connection 1-forms , curvature 2-forms , curvature tensor field (Riemann tensor field), Ricci tensor field and Ricci scalar expression which is proportional to scalar curvature.
    • We also calculate some additional quantities: Lie derivatives, exterior derivatives, interior products etc.
    Functions[f1f1[x1,x2,...,xn],f2f2[y1,y2,...,ym],...,
    fkfk[z1,z2,...,zj]]
    fk=fk[z1,z2,...,zj]-equations where fk-function identifier and zj-variables.
    Riemann[id] id - variable - corresponding identifier.
    Ricci[id]id - variable - corresponding identifier.
    RicciScalar[id]id-variable-corresponding identifier.

    Necessary functions.

Shpere - S2:

First of all we load Atlas package:
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Conformally flat metric on sphere S2:

Declare constant :
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Declare functions:
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Declear forms:
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Declare vectors:
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Declare coframe 1-forms:
Now coframe 1-forms are e1=d(x), e2=d(y). Declare frame vectors:
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Now frame vectors are . Metric declaration:
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Calculate connection 1-forms:
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Calculate curvature 2-forms:
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Riemann tensor:
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Ricci tensor:
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Ricci scalar:
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Show 1-form 2, 1:
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Verify that there is no Killing vector field among frame vector fields:
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Verify that "rotation" vector field is a Killing one:
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Calculation of volume form using Hodge operator &** :
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Some more calculations. Using interior product operator - :
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The same with metric tensor g:
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Using exterior derivative operator - d:
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As ek are coframe 1-forms and i are frame vectors then:
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"ToBasis" decomposition:
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