Riemannian cone

Definition

Given a Riemannian manifold $(M,g)$, the Riemannian cone of $M$ is defined as the manifold $M \times (0,\infty)$ equipped with the cone metric:

$t^2 g + dt^2$

Equivalently, if we actually embed the Riemannian manifold in Euclidean space $\R^N$, then this is the Riemannian manifold given by a cone in $\R^{N+1}$ whose section is $M$ (as embedded in $\R^N$).