Geodesics in non-smooth manifolds

In summary, the conversation discusses the transformation of the x-axis (y=0) into a hyperbola y=1/x, excluding the origin. The question is how to compute geodesics between two points on the hyperbola, one in the negative part and the other in the positive part. It is noted that this is not possible in differential geometry if the points have opposite signs. The possibility of using Riemann sphere and analytical continuation to solve the problem is mentioned, but the person is not familiar with this topic. It is also asked if the problem becomes solvable if all quantities are assumed to be complex and non-zero.
  • #1
mnb96
715
5
Hello,
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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  • #2
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.
 
  • #3
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?
 
Last edited:

Related to Geodesics in non-smooth manifolds

1. What is a geodesic?

A geodesic is the shortest path between two points on a curved surface or manifold. In other words, it is the path that minimizes the distance traveled between two points.

2. What are non-smooth manifolds?

Non-smooth manifolds are mathematical spaces that are not continuously differentiable. This means that they have points where the curvature changes abruptly, making it difficult to define a smooth geodesic.

3. How do non-smooth manifolds affect geodesic calculations?

Non-smooth manifolds can make it challenging to determine the shortest path between two points, as there may be multiple geodesics that are equally short. Additionally, the abrupt changes in curvature can make it difficult to calculate the geodesic equation accurately.

4. What are some real-world applications of studying geodesics in non-smooth manifolds?

Studying geodesics in non-smooth manifolds has applications in various fields, such as computer graphics, robotics, and physics. For example, understanding geodesics in non-smooth manifolds is essential for developing efficient path-planning algorithms for robots or simulating the motion of particles in complex physical systems.

5. How do scientists deal with the challenges of non-smooth manifolds when studying geodesics?

Scientists use various mathematical techniques, such as differential geometry and optimization methods, to overcome the challenges of non-smooth manifolds. They also rely on computer simulations and numerical methods to approximate geodesics and study their properties in these complex spaces.

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