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.
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3$\begingroup$ 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\$. $\endgroup$ – Igor Belegradek Dec 31 '18 at 19:09