- 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.
I never heard of Finsler spaces. Apparently they'e a superset of Riemannian space. But my question involved the latter.strangerep said:Various Finsler spaces?
Umm,... no,... your question involved non-Riemannian spaces.Paige_Turner said:I never heard of Finsler spaces. Apparently they'e a superset of Riemannian space. But my question involved the latter.
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.martinbn said:> What do you understand by non-Riemannian pseudometric space?
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 said:If > isn't okay, how should I indicate quoted text?
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.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.
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.
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.
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.
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.
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.