Manuscript XIX-A@1 · XIX-A.tex · lemma · e30d8062

Lemma: Kinematical normal component

Fourth refusal: incidence parameter and branch-normal speed › The normal-speed calculation · lines 1245–1268 · proof included or linked

XIX-A@1#lem:normal-speed, XIX-A@1#eq:normal-speed

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On a regular branch of (eq:graph), the future unit normal is

n_{s} = \frac{\partial _{U} + \nabla _{γ_{μ}} F}{1 - \nabla _{γ_{μ}} F ^{2}} .

For the deformation $V_{s} = (\partial_{s} F) \partial_{U}$ , its positive normal component is

L_{s} := - q_{cone} (n_{s}, V_{s}) = \frac{\partial _{s} F}{1 - \nabla _{γ_{μ}} F ^{2}} .

Proof. A tangent vector to the graph is $w + (d_{x} F) (w) \partial_{U}$ . Its inner product with $\partial_{U} + \nabla F$ is zero. The squared norm of the latter is $- 1 + ∣ \nabla F ∣^{2}$ . Normalization and contraction with $V_{s}$ give the formulas.

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XIX-A@1#eq:normal-speed · line 1254
$L_{s} := - q_{cone} (n_{s}, V_{s}) = \frac{\partial _{s} F}{1 - \nabla _{γ_{μ}} F ^{2}} .$
unlabeled · line 1248
$n_{s} = \frac{\partial _{U} + \nabla _{γ_{μ}} F}{1 - \nabla _{γ_{μ}} F ^{2}} .$

Lemma: Kinematical normal component

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