Problem about skew lines

Question

I have to solve this problem. There are two skew lines in the affine space. I have to find which affine isometries preserve the union of this two lines (swapping the lines could be a trasformazionee of this kind, but I am not sure if it is an affine isometry). Obviously the Identity is one of this transformation, but how can I prove that it is the only (if it is the only). I am enough sure I proved that the Identity is the only direct isometry that do this, but what about inverse isometries? Could you give me a suggestion or a piece of advice about the last part? Are there any other direction or inverse isometries which have that property? Do the number of those transformation depends on the reciprocal position of the lines? Thank you very much

Answer 1

Isometries must carry lines to lines. Within a line it can only translate and either preserve or reverse directions. However the presence of the other line prevents translation (consider how the isometry has to treat the two points of closest approach). So you can only carry each line to itself, while preserving or flipping each line (4 combinations) or carry each line to the other, each in two different possible directions, for another 4 combinations. Thus there are at most $8$ such isometries in three dimensional space. If your ambient space is higher than 3-dimensional, then isometries invoiving the other dimensions also exist (e.g., reflections and rotations preserving the 3D space generated by the two lines).

Which of these $8$ possibilities correspond to actual isometries will depend on whether the lines are perpendicular. The symmetry of the perpendicular case allows opportunities that do no exist otherwise.