Pseudometric Spaces: What Are They and Why Are They Important in Mathematics?

[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.

 

[strangerep]

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

 

[Paige_Turner]

Various Finsler spaces?

I never heard of Finsler spaces. Apparently they'e a superset of Riemannian space. But my question involved the latter.

 

[martinbn]

What do you understand by non-Riemannian pseudometric space?

 

[strangerep]

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.

 

[Paige_Turner]

> 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?

 

[robphy]

http://www.ams.org/notices/199609/chern.pdf
"Finsler Geometry Is Just Riemannian Geometry without the Quadratic Restriction"
NOTICES OF THE AMS, Sept 1996
Shiing-Shen Chernhttps://www.math.iupui.edu/~zshen/Finsler/history/rund.html
Historical Remarks on Finsler Geometry, 1959
Hanno Rund

 

[berkeman]

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.

 

[strangerep]

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.

 

[Paige_Turner]

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.