Is $dx^2+f(x)^2 dy^2$ always a complete Riemannian metric on $\mathbb{R}^2$, when $f:\mathbb{R}\to\mathbb{R}_{>0}$?

Question

I was wondering about the necessary conditions to a warped product metric $dx^2+f(x)^2 dy^2$ be a complete metric on $\mathbb{R}^2$, when $f$ is a positive function defined on $\mathbb{R}$.

Does anyone know anything about it? Is it right, or is there a easy counterexample?

I appreciate it.

Answer 1

See O’Neill, Semi-Riemannian Geometry with Applications to Relativity, chapter 7, lemma 40 (page 209) which says

If $B,F$ are complete Riemannian manifolds, then $M:=B\times_fF$ is a complete Riemannian manifold for every smooth warping function $f:B\to (0,\infty)$.

The proof uses the Hopf-Rinow theorem. He also gives a counterexample in the indefinite case, even if $B,F$ have definite metrics.

In your case, this is obviously satisfied.