Projectors - calculation of a mapping projectors (tangent and normal or horizontal and vertical)

Projectors[f]
allows one to calculate projectors of a mapping between manifolds (see Mapping).
  • f - mapping identifier
  • If mapping f is an embedding or immersion then normal and tangent projectors are calculated. If mapping f is a submersion then horizontal and vertical projectors are calculated. The corresponding rules are as follows:
  • Let mapping F:MN be declared by functions (see Mapping): where m=dim(M), n=dim(N); {x1, x2, ..xm} are local coordinates on M and {y1, y2, ..yn} are local coordinates on N.
  • - If m<n then the mapping is treated as an embedding or immersion, thus normal and tangent projectors are calculated.
  • - If n≤m then the mapping is treated as a submersion, thus horizontal and vertical projectors are calculated.
  • It should be pointed out that the calculation is only available if the actual metrics declared in both manifolds. To get more information see examples: Ricci-flat warped product.
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Domain R3
This domain is just 3-dimensional Euclidean space:
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Declare 1-forms for to use them as a coframe:
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Declare vector fields to use them as a frame:
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Declare coframe 1-forms:
Declare frame vectors:
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Declare flat metric:
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Domain S1
This domain is 1-sphere (a circle).
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Declare 1-forms for coframe:
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Declare vector fields for frame:
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Coframe declaration for the sphere:
Frame declaration for the sphere:
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Mappings
Mapping of the sphere into the Euclidean space R3:
Visualize the mapping:
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Mapping of the Euclidean space R3/Z (Z axis is excluded) into the sphere:
Metric on the sphere
After that we can calculate metric induced on the sphere by embedding:
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projectors
Calculation of the projection:
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Normal projector extraction
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Tangent projection extraction:
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Jump into the previous domain:
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Verify that there are no tangent or normal vectors among frame ones:
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Verify that "rotation" vector is tangent one:
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projectors
Calculation of the projectors:
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Vertical projector extraction:
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Horizontal projector extraction:
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Jump into R3:
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Verify that there are no tangent or normal vectors among frame ones:
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Verify that "rotation" vector is horizontal one:
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