Manuscript I-B-G@2 · I-B-G-v2.tex · section · 3b4d700f

Orientation of geometric holonomy

The prescribed host and its causal geometry › Orientation of geometric holonomy · lines 422–443

With the displayed developing map,

dev (r, φ + 2 π) = Rot (2 π α) dev (r, φ) = Rot (- μ) dev (r, φ) .

The inverse gluing generator is therefore

J_{μ} (t, X, Y, z) = (t, Rot (μ) (X, Y), z) .

Calling $J_{μ}$ the positively oriented holonomy, as in the I-B convention, amounts to choosing the corresponding gluing or inverse-meridian orientation CheungIB (:holonomy). Both generators fix $t, z$ ; they have no translational part. Reversing the chosen generator reverses every holonomy sign below. The distinction is recorded rather than hidden in the notation.

Displayed formulas in this passage

unlabeled · line 425
$dev (r, φ + 2 π) = Rot (2 π α) dev (r, φ) = Rot (- μ) dev (r, φ) .$
unlabeled · line 431
$J_{μ} (t, X, Y, z) = (t, Rot (μ) (X, Y), z) .$