Geodesics in non-smooth manifolds

[mnb96]

Hello,
I will expose a simplified version of my problem.
Let's consider the following transformation of the x-axis $$(y=0)$$ excluding the origin ($$x\neq 0$$):

$$\begin{cases} \overline{x}=x \\ \overline{y}=1/x \end{cases}$$

Now the x-axis (excluding the origin) has been transformed into an hyperbola $$y=1/x$$
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 $$x\neq 0$$) , 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!

 

[Hurkyl]

Start with a simpler question:

How can I compute geodesics between two points lying on the punctured line in a consistent way?​

The answer is that you can't, if the points have opposite sign. Differential geometry cannot tell you anything about the relationship between the two disconnected components of this space.

 

[mnb96]

Hi,
Please, note that I was forced to puncture my x-axis on the origin because 1/x is not defined on that point.

However, I somewhat recall an approach dealing with Riemann sphere and analytical continuation. In that view, many discontinuous curves in the plane (like hyperbolas) were simply projections of closed continuous curves on the Riemann sphere.

Unfortunately I am not familiar at all with that topic, so I don't know how and if it can help.
Thanks again.

EDIT:
does the problem become possible to solve if we assume that all the quantities I mentioned are complex and never 0?