Connection with torsion

This notebook illustrates how to use the Atlas package to make calculations with user defined connections.

Examples:

Functions[f1f1[x1,x2,...,xn],f2f2[y1,y2,...,ym],...,
fkfk[z1,z2,...,zj]]
fkfk(z1, z2, ..., zj) - equations where fk-function identifier and zj - variables.
Vectors[v1,v2,...,vi,...,vn]vi - vector identivier.
Forms[f1n,f2k,...,fip]fip - equations where fi - form identifier and p is a variable or an integer - the form's degree.
Coframe[id1expr1,id2expr2,...idnexprn]id - identifier for indexed variable - the coframe 1-forms n - dimension of working manifold (a variable or integer) idiexpri - equation where idi is indexed variable - coframe 1-form and expri is decomposition of the 1-form on exact 1-forms.
Frame[idj]idj -indexed variable the frame vectors

Necessary functions.

First of all we load Atlas package:
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Function declaration:
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Vector fields:
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Differential p-forms:
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Coframe 1-forms:
Frame vector fields:
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Connection definition:
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Connection[id]id - variable - connection identifier.
Curvature[idid-variable-curvature identifier.
Torsion[id]id - variable - torsion identifier.
Riemann[id]id - variable - corresponding identifier.
Ricci[id] id - variable - corresponding identifier.

Necessary functions.

Connection declaration:
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Curvature calculation:
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Result:
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Torsion calculation:
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Result:
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Curvature tensor calculation:
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Ricci tensor calculation:
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Some more simple calculations. Covariant derivatives:
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Lie derivative:
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Interior products:
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