Riemannian cone

Definition

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

$t^{2}g+d{t}^2$

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