# Is a manifold $R^{+}\times X$ with metric $dt^{2}+t^{2}\rm{g}_{X}$ complete?

The cone $$C(X)$$ over a complete manifold $$X$$ is defined as $$R^{+}\times X$$ admits the metric $$dt^{2}+t^{2}g_{X}$$. The manifold $$C(X)$$ is conformal to $$Cyl(X):=\{R^{+}\times X, dt^{2}+g_{X})$$. Following Hopf–Rinow theorem, it implies that $$Cyl(X)$$ is complete. Whether $$C(X)$$ is complete? Thanks very much.

• If $R^+$ is $(0,\infty)$ and $t$ is a function on $X$, then the answer is no, e.g., the standard $\mathbb R^n$ with origin removed is is such a warped product with $X$ being the unit sphere and $t(x)=\|x\|$. – Igor Belegradek Dec 31 '18 at 19:09