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