nLab cone (Riemannian geometry)

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

Idea

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

Examples

Spherical cones

The metric cone on the round sphere is simply Euclidean space

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

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

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

If here $G = \mathbb{Z}_n$ 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 $C(S^n/G)$ may be included. The result is the orbifold $\mathbb{R}^{n+1}\sslash G$ which is the homotopy quotient of Euclidean space by the linear $G$-action ($G$-representation).

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

$G_2$-manifolds

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

• The metric cone over complex projective 3-space carries the structure of a G₂-manifold whose Riemannian metric is invariant under the canonical Sp(2) action by left-matrix multiplication on homomogeneous coordinates in $\mathbb{H}^2 \simeq_{\mathbb{R}} \mathbb{C}^4 \to \mathbb{C}P^3$ (Byant-Salamon 89, see also Acharya-Bryant-Salamon 20).

Related concepts

• conifold

• piecewise flat spacetime

• Sasakian manifold

• conical space

• M-theory on G₂-manifolds

References

General

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)

Black brane conical singularities

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

• Bobby Acharya, Jose Figueroa-O'Farrill, Chris Hull, B. Spence, Branes at conical singularities and holography, Adv. Theor.Math. Phys.2:1249-1286, 1999 (arXiv:hep-th/9808014)

• Jose Figueroa-O'Farrill, Near-horizon geometries of supersymmetric branes, talk at SUSY 98 (arXiv:hep-th/9807149, talk slides)

• Michael Atiyah, Edward Witten, $M$-Theory dynamics on a manifold of $G_2$-holonomy, Adv. Theor. Math. Phys. 6 (2001) (arXiv:hep-th/0107177)

A suggestion that there is an analogue of AdS-CFT duality for conical bulk spacetimes and their conical singularities:

• Rong-Xin Miao, Codimension-$n$ Holography for the Cones (arXiv:2101.10031)

$G_2$-conifolds

On G₂-conifolds (G₂-manifolds with conical singularities):

Survey:

• Spiro Karigiannis, $G_2$-conifolds: A survey, 2014 (pdf)

• Mark Haskins, Exotic Einstein metrics on $S^6$ and $S^3 \times S^3$ nearly Kähler 6-manifolds and $G_2$-holonomy cones, 2016 (pdf)

Three simply connected $G_2$-cones are known: the

1. metric cone on complex projective 3-space

2. metric cone on $S^3 \times S^3$

3. metric cone on $SU(3) / (U(1) \times U(1))$

• Robert Bryant, Simon Salamon, On the construction of some complete metrics with exceptional holonomy, Duke Math. J. Volume 58, Number 3 (1989), 829-850 (euclid:euclid.dmj/1077307681)

• Gary Gibbons, Don Page, Christopher Pope, Einstein metrics on $S^3$, $\mathbb{R}^3$ and $\mathbb{R}^4$ bundles, Comm. Math. Phys. Volume 127, Number 3 (1990), 529-553 (euclid:cmp/1104180218)

More on the metric cone over complex projective 3-space as a G₂-manifold:

• Bobby Acharya, Robert Bryant, Simon Salamon, A circle quotient of a $G_2$ cone, Differential Geometry and its Applications Volume 73, December 2020, 101681 (arXiv:1910.09518, doi:10.1016/j.difgeo.2020.101681)