# Riemannian pseudometric space

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• Paige_Turner
In summary: The nonstandard system font I use (apparently) doesn't define whatever characters you use in the editor interface buttons. They look like little rectangles with a unicode hex index inside.If > isn't okay, how should I indicate quoted text?By using the PF "Reply" feature, which it looks like you are doing okay with. Just don't add an extra character into the quote -- it should be the exact quote from the other person.

#### Paige_Turner

TL;DR Summary
What is an example non-Riemannian pseudometric space that is not null or trivial?
I suspect you're not supposed to ask short questions here. Mine is in the summary.

Various Finsler spaces? (Usually, Finsler metrics are positive definite, but you can have pseudometrics too.)

wrobel
strangerep said:
Various Finsler spaces?
I never heard of Finsler spaces. Apparently they'e a superset of Riemannian space. But my question involved the latter.

What do you understand by non-Riemannian pseudometric space?

Paige_Turner said:
I never heard of Finsler spaces. Apparently they'e a superset of Riemannian space. But my question involved the latter.
Umm,... no,... your question involved non-Riemannian spaces.

A Finsler space whose fundamental function (squared) is quadratic in the velocities is Riemannian, but all other Finsler spaces are non-Riemannian.

robphy
martinbn said:
> What do you understand by non-Riemannian pseudometric space?
I have no idea. It's not my concept. Someone somewhere said that the pseudometric spaces are a proper superset of the Riemannian spaces, but I don't know the difference, and I want to learn what the difference between he two is.

META: Someone asked me not to use the > character to indicate quoted text in replies.

The nonstandard system font I use (apparently) doesn't define whatever characters you use in the editor interface buttons. They look like little rectangles with a unicode hex index inside.

If > isn't okay, how should I indicate quoted text?

Last edited:
Paige_Turner said:
If > isn't okay, how should I indicate quoted text?
By using the PF "Reply" feature, which it looks like you are doing okay with. Just don't add an extra character into the quote -- it should be the exact quote from the other person.

When you click-drag a section of another poster's text, then click "Reply", that creates the Quote Box with the other user's username and a little up-arrow that will take folks to the post that the quote came out of.

Paige_Turner said:
I have no idea. It's not my concept. Someone somewhere said that the pseudometric spaces are a proper superset of the Riemannian spaces, but I don't know the difference, and I want to learn what the difference between he two is.
Since you're a relatively new PF user, I'll explain that "someone somewhere said" is a good way to get knowlegeable people here to become disinterested in your post.

A likely-more-successful way to get better answers would have been for you to first check what an ordinary Riemannian space is (e.g., on Wikipedia), and what a pseudometric is (also on Wikipedia or other sources available by googling). Similarly, if you google for "pseudo-Riemannian space" you'll get some other references. Then, if anything is still unclear, ask a more specific question here on PF, also mentioning which specific reference sources you have already consulted.

berkeman, martinbn and robphy
I said I didn't remember where I read it.

If I had been talking about F=MA, I wouldn't have to cite that. Similarly, since this simple question is something that real physicists know, i thought that they would recognize it and just respond. If not, it isn't worth making a big deal about.

In any case, one would think you could ask a question here about something as abstract as pseudometric space without being required to say who asked it somewhere else or being told to go look up the answer.

weirdoguy and berkeman

## 1. What is a Riemannian pseudometric space?

A Riemannian pseudometric space is a mathematical concept used in the field of geometry to describe a space in which the distance between any two points is defined in terms of a pseudometric function. This function measures the "distance" between two points in a space, taking into account the curvature and other geometric properties of the space.

## 2. How is a Riemannian pseudometric space different from a Riemannian metric space?

A Riemannian pseudometric space differs from a Riemannian metric space in the sense that the distance function in a pseudometric space may not satisfy all the properties of a metric, such as the triangle inequality. This makes the notion of distance in a pseudometric space more flexible and allows for a wider range of spaces to be studied.

## 3. What are some applications of Riemannian pseudometric spaces?

Riemannian pseudometric spaces have many applications in various fields, including mathematics, physics, and computer science. In mathematics, they are used to study the geometry of curved spaces, such as Riemann surfaces and manifolds. In physics, they are used in the theory of relativity to describe the curvature of space-time. In computer science, they are used in the field of machine learning to define distance measures for clustering and classification algorithms.

## 4. How are Riemannian pseudometric spaces related to Riemannian manifolds?

Riemannian pseudometric spaces and Riemannian manifolds are closely related concepts. A Riemannian manifold is a space that locally looks like a flat space, but may have curvature at a global level. A Riemannian pseudometric space is a special type of Riemannian manifold where the distance function is defined in terms of a pseudometric. Therefore, all Riemannian pseudometric spaces can be thought of as Riemannian manifolds, but not all Riemannian manifolds are pseudometric spaces.

## 5. Can Riemannian pseudometric spaces be generalized to higher dimensions?

Yes, Riemannian pseudometric spaces can be generalized to higher dimensions, such as 3D or n-dimensional spaces. The concept of a pseudometric can also be extended to define distance functions on more general spaces, such as metric spaces and topological spaces. However, the study of Riemannian pseudometric spaces is primarily focused on 2-dimensional spaces, as they have many applications and are easier to visualize and understand.