Pullback - Pullback of a [0, k] tensor field under a mapping

Pullback[expr, mapId]
allows one to calculate pullback of a covariant tensor field under a mapping.
  • mapId - variable - the mapping identifier i.e. mapId : dom1 ---> dom2 expr - expression - a tensor expression which has to be restricted
  • The pullback is linear operation defined on [0,k] tensors only. The definition is as follows.
  • Let M and N be manifolds of dimensions m=dim(M), n=dim(N). Let F be mapping between the manifolds: F:MN defined by functions: where {x1, x2, ..xm} are local coordinates on M and {y1, y2, ..yn} are local coordinates on N (in some domains).
  • For any [0,1] tensor field T on N the pullback of T under F is tensor field =Pullback[T, F] on M with components Pullback[((T1)(T2)), F]=Pullback[T1, F]Pullback[T2, F] in local coordinates.
  • For tensor product of any [0, k] tensor fields on N the following formula takes place:
  • The formulas considered above completely define the linear pullback operator Pullback.
  • According to the definition it is necessary to calculate the pullbacks on the domain M. Use Domain procedure to jump on M manifold if needed.
The following example shows how the pullback operator can be used.
Let M be 2-dimentional sphere S2 and N be 3-dimensional Euclidean space R3. Let F:MN be standard embedding of sphere S2 into R3.
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Declare 1-forms ej and uk for corresponding coframes:
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Declare vectors for corresponding frames:
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Declare Euclidean space - R3:
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Declare coframe on R3:
Declare frame on R3:
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Declare metric on R3 (standard flat metric):
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Declare sphere - S2:
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Declare coframe on S2:
Declare frame on S2:
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Declare definite mapping F:S2R3:
Visualize the mapping:
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Verify that we are on the sphere:
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Calculate metric induced on the sphere using pullback operator:
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One can calculate pullback of any [0,k] tensor field on R3 under the mapping: pullback of coframe 1-forms:
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pullback of 0-forms (scalars):
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pullback of "rotation" 1-form:
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pullback of tensor product d(x)d(z):
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pullback of exterior product d(x)^d(y):
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Some more examples:
Declare abstract mapping between S2 and R3:
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pullback of exterior product d(x)^d(y) under abstract mapping :
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pullback of coframe 1-forms
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Who is who?
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