For other, related, concepts of a similar name see at cone.
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”.
The metric cone on the round sphere is simply Euclidean space
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 $S^n$, 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.
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).
Discussion in the context of 3-manifolds and orbifolds:
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)
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
metric cone on $S^3 \times S^3$
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:
Last revised on October 14, 2020 at 09:09:52. See the history of this page for a list of all contributions to it.