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EDIT: As pointed out by Pete below, it seems I misunderstood the question, so what I write below is not relevant. Apologies!

This is not the general answer, but in riemannian geometry there is a notion of cone. If $(M,g)$ is a riemannian manifold, then its metric cone is $\mathbb{R}^+ \times M$, with $\mathbb{R}^+$ the positive real half-line parametrised by $r>0$, with metric $$dr^2 + r^2 g$$ The best example is of course $(M,g)$ the unit sphere in $\mathbb{R}^n$ and its cone is then $\mathbb{R}^n \setminus \lbrace 0\rbrace$. In this case (and in this case only) the metric extends smoothly to the origin, but in general the apex of the cone is singular.

This is used as a local model for conical (!) singularities and there is a nice interplay between the geometry of $M$ and that of its cone. The most dramatic use of the cone I know is that it turns the problem of determining which riemannian spin manifolds admit real Killing spinors into a holonomy problem, namely the determination of which metric cones admit parallel spinors.

Some of this generalises to the pseudo-riemannian setting; although this is perhaps not as useful as in the riaemannian setting as the holonomy classification in indefinite signatures (except for lorentzian) is still lacking.

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This is not the general answer, but in riemannian geometry there is a notion of cone. If $(M,g)$ is a riemannian manifold, then its metric cone is $\mathbb{R}^+ \times M$, with $\mathbb{R}^+$ the positive real half-line parametrised by $r>0$, with metric $$dr^2 + r^2 g$$ The best example is of course $(M,g)$ the unit sphere in $\mathbb{R}^n$ and its cone is then $\mathbb{R}^n \setminus \lbrace 0\rbrace$. In this case (and in this case only) the metric extends smoothly to the origin, but in general the apex of the cone is singular.

This is used as a local model for conical (!) singularities and there is a nice interplay between the geometry of $M$ and that of its cone. The most dramatic use of the cone I know is that it turns the problem of determining which riemannian spin manifolds admit real Killing spinors into a holonomy problem, namely the determination of which metric cones admit parallel spinors.

Some of this generalises to the pseudo-riemannian setting; although this is perhaps not as useful as in the riaemannian setting as the holonomy classification in indefinite signatures (except for lorentzian) is still lacking.