EXISTENCE OF QUASI-DOUBLE LINES OF THE PAIR OF THREE DIMENSIONAL DISTRIBUTIONS IN THE PARTIAL MAPPING OF SPACE Ε_6
DOI:
https://doi.org/10.52754/16948645_2024_2(5)_20Keywords:
Euclidean space, Frenet’s frame, net of Frenet, partial mapping, distribution, quasi-double line of a pair of distributionsAbstract
A family of smooth lines given in the domain so that through each point passes one line of given family. A movable frame is chosen so that it was Frenet’s frame for the line of the given family. The integral lines of the coordinate vectors fields of this frame form a Frenet’s net. On a tangent to the line of this net a point is defined in an invariant way. When the point moves in the domain , the point describes it's domain In this way we get a partial mapping such, that .
It is considered the three dimensional distributions and .
Definition. Lines and are called quasi-double lines of the pair of distributions , If the tangent to line at the point and the tangent to line at point belong to the same three dimensional space, spanned by vectors .
Necessary and sufficient conditions have been proven for lines and to be quasi-double lines of the pair of distributions , in the partial mapping .
References
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