This domain is just 3-dimensional Euclidean space:
Declare 1-forms for to use them as a coframe:
Declare vector fields to use them as a frame:
This domain is 1-sphere (a circle).
Declare 1-forms for coframe:
Declare vector fields for frame:
Coframe declaration for the sphere:
Frame declaration for the sphere:
Mapping of the sphere into the Euclidean space
R^{3}:
Mapping of the Euclidean space
R^{3}/Z (Z axis is excluded) into the sphere:
After that we can calculate metric induced on the sphere by
embedding:
Calculation of the projection:
Normal projector extraction
Tangent projection extraction:
Jump into the previous domain:
Verify that there are no tangent or normal vectors among frame ones:
Verify that "rotation" vector
is tangent one:
Calculation of the projectors:
Vertical projector extraction:
Horizontal projector extraction:
Verify that there are no tangent or normal vectors among frame ones:
Verify that "rotation" vector
is horizontal one: