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.)