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cone (Riemannian geometry)

Contents

For other, related, concepts of a similar name see at cone.

Contents

Idea

In (pseudo-)Riemannian geometry, a cone is a part of a (pseudo-)Riemannian manifold where the metric tensor is locally of the form ds 2=dr 2+r 2ds 1 2d s^2 = d r^2 + r^2 d s^2_1. The point that would correspond to r=0r = 0 is the “conical singularity”.

Examples

Spherical cones

The metric cone on the round sphere is simply Euclidean space

C(S n) n+1{0} C(S^n) \simeq \mathbb{R}^{n+1} \setminus \{0\}

and hence may in fact be continued non-singularly also at the cone tip.

For GG a finite group with a free action on the round sphere S nS^n, the quotient space S n/GS^n/G exists as a Riemannian manifold. The metric cone C(S n/G)C(S^n/G) on this is singular at the origin as soon as GG is not the trivial group.

If here G= nG = \mathbb{Z}_n is a cyclic group one says that this cone is obtained from flat Euclidean space by introducing a “deficit angle”.

If one passes beyond smooth manifolds to orbifolds, then the cone tip in C(S n/G)C(S^n/G) may be included. The result is the orbifold n+1G\mathbb{R}^{n+1}\sslash G which is the homotopy quotient of Euclidean space by the linear GG-action (GG-representation).

Such conical singularities appear for instance in the far-horizon geometry of BPS black branes. Special cases are ADE-singularities.

G 2G_2-manifolds

Three examples of cones admitted by simply-connected G2-manifolds seem to be known: cones on P 3\mathbb{C}P^3, SU(3)/U(1)×U(1)SU(3)/U(1)\times U(1) and S 3×S 3S^3 \times S^3. (Atiyah-Witten 01)

References

Discussion in the context of 3-manifolds and orbifolds:

  • Daryl Cooper, Craig Hodgson, Steve Kerckhoff, Three-dimensional Orbifolds and Cone-Manifolds, MSJ Memoirs Volume 5, 2000 (pdf, euclid:1389985812)

Discussion of supergravity black brane-solutions at conical singularities (cone branes) includes the following (see also at far-horizon geometry)

Last revised on November 13, 2018 at 03:59:33. See the history of this page for a list of all contributions to it.