nLab
cone (Riemannian geometry)
Contents
For other, related, concepts of a similar name see at cone .

Context
Riemannian geometry
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 $d s^2 = d r^2 + r^2 d s^2_1$ . 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 $S^n$ , 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 )

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 )

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 )

Last revised on November 13, 2018 at 03:59:33.
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