TITLE: Geodesic completeness of manifolds with "warped" metrics
QUESTION [5 upvotes]: Let $(M,g)$ be a geodesically complete Riemannian manifold, and let $f: M\to \mathbb{R}^+$ be a smooth, bounded strictly positive function on $M$; i.e., there exist $L,A\in \mathbb{R}^+$ such that $0<L<f(m)<A<\infty$ for all $m\in M$. Since $f$ is positive, $f^2g$ defines a metric on $M$. Since $(M,g)$ is geodesically complete, is $(M, f^2g)$ also geodesically complete?
(Note: This question was inspired as a generalization of another question posed recently about the geodesic completeness of $\mathbb{R}\times \mathbb{S}^{n-1}$, given a warped metric $ds^2=dr^2+\psi^2d\theta^2$ where $dr$ and $d\theta$ are the metrics from $\mathbb{R}$ and $\mathbb{S}^{n-1}$ respectively and $\psi$ is a positive function on $\mathbb{R}\times \mathbb{S}^{n-1}$.)

REPLY [2 votes]: [This is an answer to the original question]
$\newcommand{\R}{\mathbb{R}}$
Let $M := \R$ with the standard metric $g$, and $f(x) := e^{-x^2}$. Then the whole line $M$ has finite length if given the metric $f^2 g$, which shows it cannot be geodesically complete.