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I will expose a simplified version of my problem.

Let's consider the following transformation of the x-axis [tex](y=0)[/tex]excluding the origin([tex]x\neq 0[/tex]):

[tex]\begin{cases} \overline{x}=x \\ \overline{y}=1/x \end{cases}[/tex]

Now the x-axis (excluding the origin) has been transformed into an hyperbola [tex]y=1/x[/tex]

My question is: how can I compute geodesics between two points lying on the hyperbola in a consistent way?

In other words, in the original x-axis (with [tex]x\neq 0[/tex]) , we can compute distances between a point in the negative part and another point in the positive part by simply "filling the gap" including also the zero. Now, Is it possible to compute geodesics between two points on the hyperbola, one in the negative part, and the other in the positive part?

Thanks!

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# Geodesics in non-smooth manifolds

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