I am unsure as to how I can correctly answer this question: Show that there is a one to one correspondence between the set of all parallel translations and the set of all vectors in space
2026-04-07 12:34:36.1775565276
Show that there is a one to one correspondence between the set of all parallel translations and the set of all vectors in space
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You can say that any vector $v$ defines a parallel translation via $x \mapsto x +v$. Conversely, choose an arbitrary parallel translation according to your favourite definition and show that it corresponds to $x \mapsto x +w$ for some vector $w$, this gives you the inverse of the one-to-one correspondence that you are looking for.
Hope this helps.