Manuscript VII-F-ST@1 · VII-F-ST.tex · section · 9eee7a44

Adapted affine charts and framed peripheral holonomy

Adapted affine charts and framed peripheral holonomy · lines 720–765

VII-F-ST@1#eq:chart-table

We now verify the full affine certificate, not merely its Lorentz part. The source chart pattern is VII-F's finite table CheungVIIF (:holonomy).

In $(t, ξ, η)$ , put

B_{h} = 3 - 2 - 2 - 2 21 - 2 12, B_{x} = 5/4 - 3/4 0 - 3/4 5/4 0 001, B_{y} = 5/4 0 - 3/4 010 - 3/4 0 5/4 .

Let $O_{i} = diag (1, R (iπ /2))$ . Extend all matrices by the identity on $ζ$ . With $e_{0} = (1, 0, 0, 0)^{T}$ , define

f V_{i} U_{i}^{-} U_{i}^{+} Q_{i}, exterior A_{f} O_{i} B_{h} O_{i}^{- 1} - 3 s e_{0} O_{i} B_{y} O_{i}^{- 1} - (5 s /2) e_{0} O_{i} B_{x} O_{i}^{- 1} - (5 s /2) e_{0} id face equation after adaptation t^{'} = 0 t^{'} = 0 t^{'} = 0 t^{'} = s + 2

An entry such as $L - a e_{0}$ denotes the affine map $u \mapsto Lu - a e_{0}$ .

Displayed formulas in this passage

VII-F-ST@1#eq:chart-table · line 753
$f V_{i} U_{i}^{-} U_{i}^{+} Q_{i}, exterior A_{f} O_{i} B_{h} O_{i}^{- 1} - 3 s e_{0} O_{i} B_{y} O_{i}^{- 1} - (5 s /2) e_{0} O_{i} B_{x} O_{i}^{- 1} - (5 s /2) e_{0} id face equation after adaptation t^{'} = 0 t^{'} = 0 t^{'} = 0 t^{'} = s + 2$
unlabeled · line 728
$B_{h} = 3 - 2 - 2 - 2 21 - 2 12, B_{x} = 5/4 - 3/4 0 - 3/4 5/4 0 001, B_{y} = 5/4 0 - 3/4 010 - 3/4 0 5/4 .$