1
$\begingroup$

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.

$\endgroup$
  • 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

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.