Let $M$ be a noncompact $C^\infty$ manifold, let $X$ be a complete $C^\infty$ vector field on $M$, and take $f\in C^\infty\big(M;(0,\infty)\big)$ a strictly positive function.


Question: Does anyone know sufficient conditions on the function $f$ implying the completeness of the vector field $fX$ ?


(When $M$ is compact, the vector field $fX$ is complete and has the
same integral curves as the vector field $X$, cf. Chapter 2, Section 2
of the book of Ergodic Theory of Cornfeld, Fomin and Sinai.)



completeness of a vector field fX, with X complete and f>0
If $f$ is bounded, then the rate at which one travels along the integral curves of $X$ is only increased by a bounded factor, so it still takes infinite time to get all the way along each integral curve,so $fX$ is complete.


