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Martin Sleziak
Martin Sleziak
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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 riemannianRiemannian geometry there is a notion of cone. If $(M,g)$ is a riemannianRiemannian 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 riemannianRiemannian 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 riaemannianRiemannian setting as the holonomy classification in indefinite signatures (except for lorentzianLorentzian) is still lacking.

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.

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 Riemannian setting as the holonomy classification in indefinite signatures (except for Lorentzian) is still lacking.

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José Figueroa-O'Farrill
José Figueroa-O'Farrill
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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.

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.

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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José Figueroa-O'Farrill
José Figueroa-O'Farrill
  • 33.3k
  • 4
  • 105
  • 183

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.