Here's something really weird. As can be read in Pressley's "Elementary Differential Geometry":(adsbygoogle = window.adsbygoogle || []).push({});

Proposition 1.3: Any reparametrization of a regular curve is regular.

And 4 pages later:

Exemple 1.8: For the parametrization [itex]\gamma(t)=(t,t^2)[/itex] of the parabola y=x², [itex]\dot{\gamma}[/itex] is never 0 so [itex]\gamma[/itex] is regular. But [itex]\tilde{\gamma}(t)=(t^3,t^6)[/itex] is also a parametrization of the same parabila. This time, [itex]\dot{\tilde{\gamma}}=(3t^2,6t^5)[/itex] and this is zero when t=0, so [itex]\tilde{\gamma}[/itex] is not regular.

Just to makesurethat [itex]\tilde{\gamma}[/itex] is a reparametrization of [itex]\gamma[/itex], consider the reparametrization map [itex]\phi:(-\infty,+\infty)\rightarrow (-\infty,+\infty)[/itex] define by [itex]\phi(t)=t^3[/itex]. Then [itex]\phi[/itex] is a smooth bijection with a smooth inverse such that [itex]\gamma \circ \phi = (\phi(t),\phi(t)^2)=(t^3,t^6)= \tilde{\gamma}[/itex], so [itex]\tilde{\gamma}[/itex] is really a reparametrization of [itex]\gamma[/itex] but it is not regular, contradicting proposition 1.3.

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# Something weird! (diff geom)

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