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Let $X$ and $Y$ be complete commuting vector fields on a (semi-) Riemannian manifold.
Denote by $\pi X$ the component of $X$ orthogonal to $Y$, i.e. $\pi X = X - \frac{\langle X,Y\rangle}{\langle Y,Y\rangle} Y$.

Can we say anything about whether $\pi X$ is complete?

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    $\begingroup$ What does it mean complete for you? $\endgroup$
    – Federico Fallucca
    Commented Apr 8, 2022 at 11:42
  • $\begingroup$ With completeness I mean that any maximal integral curve maps from $\mathbb{R}$. $\endgroup$
    – Oliver
    Commented Apr 8, 2022 at 11:49
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    $\begingroup$ So under the stronger assumtptions of $X$ and $Y$ Killing, I actually derived what I wanted, that then $X$ is complete if and only if $\pi X$ is. Indeed $\frac{\langle X,Y\rangle}{\langle Y,Y\rangle} Y$ is still complete and commutes with X, and then the sum is complete because the flow of the sum of two commuting complete vector fields is just the concatenation of their global flows. $\endgroup$
    – Oliver
    Commented Apr 10, 2022 at 22:00

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