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## Main Question or Discussion Point

I have read statements like "assume that there exists a killingvector ##\xi## that makes it possible to compactify the space in it's direction." It's not hard to find examples of compact "directions" with a corresponding killing vector (rotational direction on a sphere), but there are also examples of killing vectors pointing in a non-compact direction (translations in ##\mathbb{R}^n)##. Is there however a requirement that there exists a killing vector in the given direction in order for it to be "compactable"? I.e. does the non-existence of a killing-vector imply that the "direction" is non-compactable? If so why?