Manuscript XIX-C@1 · XIX-C.tex · lemma · 98b067d9

Lemma: Normal component and reparametrization

Parameter, code, deformation rate, and lapse › Normal speed of a prescribed graph · lines 594–620 · proof included or linked

XIX-C@1#lem:normal, XIX-C@1#eq:normal, XIX-C@1#eq:retiming

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The future unit normal and the normal deformation component are

n_{s} = \frac{\partial _{U} + \nabla _{γ_{μ}} F}{1 - \nabla _{γ_{μ}} F ^{2}}, L_{s} = - q_{μ} (n_{s}, (\partial_{s} F) \partial_{U}) = \frac{\partial _{s} F}{1 - \nabla _{γ_{μ}} F ^{2}} .

If $r = f (s)$ , $f^{'} > 0$ , describes the same unparametrized slices, then

L_{r} = \frac{L _{s}}{f ^{'} ( s )} .

Proof. For a graph tangent $w + d_{x} F (w) \partial_{U}$ , its pairing with $\partial_{U} + \nabla F$ vanishes. The squared norm of that vector is $- 1 + ∣ \nabla F ∣^{2}$ . Normalization gives $n_{s}$ , and contraction gives $L_{s}$ . The chain rule gives $\partial_{r} F = \partial_{s} F / f^{'} (s)$ , proving the final formula.

Displayed formulas in this passage

XIX-C@1#eq:normal · line 597
$n_{s} = \frac{\partial _{U} + \nabla _{γ_{μ}} F}{1 - \nabla _{γ_{μ}} F ^{2}}, L_{s} = - q_{μ} (n_{s}, (\partial_{s} F) \partial_{U}) = \frac{\partial _{s} F}{1 - \nabla _{γ_{μ}} F ^{2}} .$
XIX-C@1#eq:retiming · line 608
$L_{r} = \frac{L _{s}}{f ^{'} ( s )} .$

Lemma: Normal component and reparametrization

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