- #1
LagrangeEuler
- 717
- 20
What's the difference between Euclidean and Riemann space? As far as I know ##\mathbb{R}^n## is Euclidean space.
Shyan said:Let's replace the word "space" with "manifold" because its more general.
A Riemannian manifold is a manifold having a positive definite metric.
A Euclidean manifold is a special case of a Riemannian manifold where the positive definite metric is a Euclidean metric
In fact I was considering the "space" in the OP to mean 3-dimensional Euclidean manifold!jgens said:Unless by space you mean something like vector spaces, it's actually the other way around. The restriction to manifolds is necessary, however, since they are precisely the spaces on which Riemannian metrics are defined.
.
I was starting to feel that way too,because the wikipedia page on Riemannian manifolds were defining Riemannian metrics somehow that I couldn't relate it to the definition of metric in metric spaces!jgens said:They are technically different kinds of metrics. The metrics you learn about when studying metric spaces are very different than Riemannian metrics.
LagrangeEuler said:What's the difference between Euclidean and Riemann space? As far as I know ##\mathbb{R}^n## is Euclidean space.
Euclidean space is a flat, three-dimensional space where the laws of Euclidean geometry apply. It is the familiar space we experience in our everyday lives. Riemann space, on the other hand, is a non-Euclidean space where the laws of non-Euclidean geometry apply. It is a curved, multi-dimensional space that is used in physics to describe the universe.
Euclidean space is used in classical mechanics to describe the motion of particles and objects, while Riemann space is used in general relativity to describe the curvature of spacetime. Riemann space is also used in other areas of physics, such as quantum mechanics, where it is used to describe the geometry of multi-dimensional spaces.
Riemann space is an extension of Euclidean space, meaning that Euclidean space can be seen as a special case of Riemann space. In Riemann space, the laws of Euclidean geometry still hold locally, but the overall geometry of the space is curved.
In Euclidean space, parallel lines never intersect and are always equidistant from each other. In Riemann space, parallel lines can intersect and their distance from each other can vary depending on the curvature of the space. This is a consequence of the non-Euclidean geometry in Riemann space.
Euclidean space can be easily visualized as it is the space we experience in our daily lives. However, Riemann space is a multi-dimensional space that cannot be directly visualized. Mathematicians and physicists use mathematical tools and models to represent and study Riemann space.