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Paige_Turner
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- 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 geometry and analysis to describe a space where distances between points are measured using a pseudometric, which is a function that satisfies all the properties of a metric except for the requirement that the distance between any two distinct points must be positive. This allows for the study of spaces with negative or zero distances between points.
The key properties of a Riemannian pseudometric space include symmetry, non-negativity, and the triangle inequality. Symmetry means that the distance between any two points is the same regardless of the order in which they are considered. Non-negativity means that the distance between any two points is either zero or a positive value. The triangle inequality states that the distance between two points is always less than or equal to the sum of the distances between those points and a third point.
A Riemannian pseudometric space differs from a metric space in that the distance between two points can be zero or negative, whereas in a metric space, the distance between any two distinct points must be positive. Additionally, a Riemannian pseudometric space allows for the study of spaces with non-Euclidean geometries, while metric spaces are limited to Euclidean geometries.
Riemannian pseudometric spaces have applications in various fields such as physics, computer science, and economics. In physics, they are used to describe the curvature of spacetime in Einstein's theory of general relativity. In computer science, they are used in machine learning algorithms to measure the similarity between data points. In economics, they are used to model the relationships between different economic variables.
Some common techniques used to study Riemannian pseudometric spaces include differential geometry, topology, and analysis. Differential geometry is used to study the geometric properties of these spaces, while topology is used to study their global structure. Analysis is used to study the behavior of functions on these spaces, such as the concept of differentiability.