Manuscript VII-F@1 · VII-F.tex · proposition · 2d81c03f

Proposition: Framed affine peripheral certificate

Adapted affine charts and exact holonomy › A common framed primitive meridian · lines 1176–1209 · proof included or linked

VII-F@1#prop:holonomy

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For every $s \in [- 1, 1]$ ,

Hol (γ_{s}) = J .

The collar meridian and the core meridian are primitive and positively oriented. The same identity holds for both event germs at $s = 0$ .

Proof. The angular lift of either meridian increases by exactly $L$ , so its winding number is one.

For the core meridian, write $A_{i} = A_{V_{i}}$ . Starting in the $V_{0}$ chart, the three-face product is

(A_{0} J A_{2}^{- 1}) (A_{2} A_{1}^{- 1}) (A_{1} A_{0}^{- 1}) = A_{0} J A_{0}^{- 1} .

The path from the collar reference chart into $V_{0}$ contributes $A_{0}$ , and the return path contributes $A_{0}^{- 1}$ . The based product is therefore

A_{0}^{- 1} (A_{0} J A_{0}^{- 1}) A_{0} = J .

This is an equality in the full affine group; the translations cancel as well as the Lorentz factors.

At $s = 0$ , the core meridian avoids every collapsed cell. Its charts and transitions are common to the two germs, giving the same identity.

Displayed formulas in this passage

unlabeled · line 1179
$Hol (γ_{s}) = J .$
unlabeled · line 1192
$(A_{0} J A_{2}^{- 1}) (A_{2} A_{1}^{- 1}) (A_{1} A_{0}^{- 1}) = A_{0} J A_{0}^{- 1} .$
unlabeled · line 1200
$A_{0}^{- 1} (A_{0} J A_{0}^{- 1}) A_{0} = J .$

Nearby in this section

section: A common framed primitive meridian