Manuscript III-B@1 · III-B-v1.tex · section · d2c98f62

The conical product

Prescribed geometry and the local analytical domain › The conical product · lines 317–342

III-B@1#eq:metric

Fix

0 < μ < 2 π, α = 1 - \frac{μ}{2 π}, c > 0.

Let

C_{μ}^{\circ} = (0, \infty) \times (R /2 π Z),

and prescribe

M = R_{T} \times C_{μ}^{\circ} \times R_{z}, g ˉ = - c^{2} d T^{2} + d r^{2} + α^{2} r^{2} d ϕ^{2} + d z^{2} .

The metric completion has singular worldsheet

W = R_{T} \times {o} \times R_{z},

where $o$ is the cone apex. We use only the smooth regular part. The background and its unit lapse are supplied data, as in I-B CheungIB.

Displayed formulas in this passage

III-B@1#eq:metric · line 329
$M = R_{T} \times C_{μ}^{\circ} \times R_{z}, g ˉ = - c^{2} d T^{2} + d r^{2} + α^{2} r^{2} d ϕ^{2} + d z^{2} .$
unlabeled · line 320
$0 < μ < 2 π, α = 1 - \frac{μ}{2 π}, c > 0.$
unlabeled · line 325
$C_{μ}^{\circ} = (0, \infty) \times (R /2 π Z),$
unlabeled · line 336
$W = R_{T} \times {o} \times R_{z},$