Riemannian cone

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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).