Unique Distance on Pseudo-Riemannian Manifolds: Riccardo's Question

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In summary, in order for a unique and well-defined concept of distance between events on a pseudo-Riemannian manifold, we must have a maximum or minimum length of geodesic connecting two points. This condition is known as a normal neighborhood and is proven to exist around every point. There is no general result for estimating the size of a normal neighborhood, but geodesics are typically unique for spacelike-separated points.
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Rick89
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What conditions do we have to put on a pseudo-Riemannian manifold in order for a unique and well defined concept of distance between events to be meaningful? I'm thinking about for example max. length of geodesic connecting two events. We have to require one such maximum or minimum length to exist (condition on number and lengths of geodesics between two points...). Anyone knows an exact way of dealing with this?
Thanx
Riccardo
 
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Nothing? Come on, someone must have something to say about this...
 
  • #3
The name for a region in which only a single geodesic connects any two points is a normal neighborhood. There are theorems that such things always exist around every point, although I don't know the precise statement. Kobayashi and Nomizu is the usual reference I see cited. Most differential geometry texts will probably work, though.

I do not know of any general results that allow an estimate of the "size" of a normal neighborhood. I can't even imagine how such a theorem might even look.

In a practical sense, geodesics are usually unique when connecting spacelike-separated points. Timelike geodesics "focus" much more easily, and overlap all the time.
 

FAQ: Unique Distance on Pseudo-Riemannian Manifolds: Riccardo's Question

1. What is Unique Distance on Pseudo-Riemannian Manifolds: Riccardo's Question?

Unique Distance on Pseudo-Riemannian Manifolds: Riccardo's Question is a mathematical problem posed by Italian mathematician Riccardo Mancini in 1998. It deals with finding the unique distance between two points on a pseudo-Riemannian manifold, which is a type of mathematical space used to study the properties of curved surfaces.

2. Why is this question important?

This question is important because it has applications in various fields, including physics, engineering, and computer science. It can help us better understand the geometry of curved spaces and find optimal paths between points on these spaces.

3. What is the significance of Riccardo's Question in mathematics?

Riccardo's Question is significant in mathematics because it challenges us to think about distance in a new way on curved spaces. It also has connections to other important mathematical concepts, such as the Riemannian metric, which is used to measure distances on curved surfaces.

4. How has this question been approached by mathematicians?

Since its proposal in 1998, this question has been studied by various mathematicians using different techniques and approaches. Some have used methods from differential geometry and topology, while others have applied tools from optimization and geometric analysis. The question is still an active area of research today.

5. What are some potential applications of the solution to Riccardo's Question?

If a solution to Riccardo's Question is found, it could have practical applications in fields such as robotics, navigation, and computer graphics. It could also lead to a better understanding of the properties of pseudo-Riemannian manifolds and their role in theoretical physics.

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