# nLab cone (Riemannian geometry)

Contents

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

### Context

#### Riemannian geometry

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.

### 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)

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

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 $SU(3) / (U(1) \times U(1))$

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