Manuscript En03-A@1 · En03-A.tex · lemma · 5398614e

Lemma: Developing holonomy and tangent transport

External conical geometry and certified local moves › Tangent holonomy · lines 1278–1305 · proof included or linked

En03-A@1#lem:soldering, En03-A@1#eq:affine-development

Suppose

D \circ T_{γ}^{- 1} = h_{γ} \circ D, h_{γ} (y) = L_{γ} y + b_{γ} .

With this convention, parallel transport around $γ$ , in the starting developed frame, has matrix $L_{γ}$ .

Proof. Lift the loop from $x$ to $T_{γ} x$ . Flat developed transport keeps the Minkowski vector constant:

d D_{T_{γ} x} (v^{'}) = d D_{x} (v) .

Endpoint identification uses $d T_{γ}^{- 1}$ . Differentiating eq:affine-development gives

d D_{x} d T_{γ}^{- 1} (v^{'}) = L_{γ} d D_{T_{γ} x} (v^{'}) = L_{γ} d D_{x} (v) .

Displayed formulas in this passage

En03-A@1#eq:affine-development · line 1281
$D \circ T_{γ}^{- 1} = h_{γ} \circ D, h_{γ} (y) = L_{γ} y + b_{γ} .$
unlabeled · line 1294
$d D_{T_{γ} x} (v^{'}) = d D_{x} (v) .$
unlabeled · line 1300
$d D_{x} d T_{γ}^{- 1} (v^{'}) = L_{γ} d D_{T_{γ} x} (v^{'}) = L_{γ} d D_{x} (v) .$