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I have read these questions about smooth arc connected and arc connected manifolds:

Smooth curves on a path connected smooth manifold

Connectivity, Path Connectivity and Differentiability

But I'm working with a specific problem and having those arcs regular would be pretty nice. Do someone know if we could take the arcs regular or where to find some information about it?

Ted Shifrin
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Valere
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  • How do the linked questions not answer your question in detail? – Ted Shifrin Mar 02 '23 at 17:19
  • @TedShifrin maybe it is not a wide used name but by regular curve I mean that if $\alpha :I \rightarrow \mathbb{R}^n$ is a curve and $I$ an interval, then $\alpha '(t) \neq 0 \forall t \in I$. I need the curve to be regular in order to use parametrization by arc length. In the posts they tell how to build a smooth curve, but nothing is said about it being regular. The use of the same word for different meanings depending on the branch is driving me a bit crazy – Valere Mar 02 '23 at 17:27
  • Ah, right, so you're working with actual parametrized curves, rather than just their images. I think the usual smoothing technique of convolving with an approximation of the identity will work fine, as it will give a smooth function that is $\epsilon$-close in the $C^1$ (in fact, even better) topology. Since the original function has derivative of norm $1$, except, say, at $t=t_0$, the smoothed version will have derivative of norm at least $1-\epsilon$ everywhere. – Ted Shifrin Mar 02 '23 at 17:42

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