Riemannian Geometry: What is It?

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In summary, the conversation is about Non-Euclidean geometry, specifically Hyperbolic geometry, which is a dramatic shift from the commonly accepted Euclidean geometry. This shift is mostly psychological as reality appears to obey Euclidean geometry in everyday life. The conversation also mentions Lobachevskii, the Russian mathematician who first developed this geometry, and the different models used to visualize it, such as the disk model and the half-plane model. Finally, there is a discussion about the generalization of these models to higher dimensions.
  • #1
Ebolamonk3y
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I heard one of my friends talking about a math or a geometry invented by some famous Russian guy... It goes something like...

2 parallel lines come from the same point... And that's the base of everything else... Some crazy stuff! I been thinking about this... You can be standing on a road, and you have that vantage point effect where the horizon kind of fades off to the distance... But its still 2 parallel lines, the street/road... Yet to your view it looks like its coming from a point...

So is this geometry a mere shift of POV or something vastly different?
 
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  • #2
Ebolamonk3y said:
I heard one of my friends talking about a math or a geometry invented by some famous Russian guy... It goes something like...

2 parallel lines come from the same point... And that's the base of everything else... Some crazy stuff! I been thinking about this... You can be standing on a road, and you have that vantage point effect where the horizon kind of fades off to the distance... But its still 2 parallel lines, the street/road... Yet to your view it looks like its coming from a point...

So is this geometry a mere shift of POV or something vastly different?

You seem to be referring to Non-Euclidean geometry. For example in Hyperbolic geometry you can have multiple parallel lines passing through the same point.

It's a somewhat dramatic shift in thinking. At the time is was developed, it was very dramatic; Euclidean geometry was generally accepted as "real" geometry. The drama is mostly just psycological...from an everyday point of view, reality appears to obey Euclidean geometry, so the idea of using a different type of geometry can be disturbing.
 
  • #3
multiple parallel lines parallel to each other? That would be what I am talking about... A is parallel to B... A and B come from the same origin... Hehe, what would be better is if A and B ended at the same point as well! Squeezing space. :)
 
  • #4
I'm no expert on higher mathematics but anyway:
Just think of Longitudes. Don't they all intersect at the North Pole (and South Pole) even though they're parellel?
 
  • #5
lattitudes aren't geodesics, and longitudes aren't parallel. .

For the record, the simplest model of hyperbolic geometry is the unit disc in the plane, where the geodesics are circles that meet the edge of the disc at right angles. You can easily imagine there being an infinite number of geodesics passing through a given point and parallel to another given geodesic.
 
  • #6
The "Russian Guy" was Lobachevskii. And the model Matt Grime is talking about is "Euler's disk model" (although, personally, I think Euler's "half plane model" is simpler).
 
  • #7
Yeah! I think that's the guy! What is this math called?
 
  • #8
Ebolamonk3y said:
Yeah! I think that's the guy! What is this math called?
it is called hard
 
  • #9
LOL, ahahahah
 
  • #10
The very first reply told you that it was "non-Euclidean geometry". There is a classic book called "non-Euclidean" geometry, written by Bonola, that has been reprinted by Dover.
 
  • #11
HallsofIvy said:
The "Russian Guy" was Lobachevskii. And the model Matt Grime is talking about is "Euler's disk model" (although, personally, I think Euler's "half plane model" is simpler).

Genuine query, and explanation of why I prefer the disk model:

I know the disk model generalizes to higher dimensions, is there a generalization for the half plane? I can think of two possibilities for 3-dim space, and I guess the one where geodesics are hemispheres and planes orthogonal to the x-y plane (where I take the model to be the triples (x,y,z) in R^3 with z>0) is the 'correct' one.
 
  • #12
The disk and half-plane models are really the same thing. Imagine a small section of the disk, next to the bounding circle is "blown up" (expanded). If you make it big enough, the bounding circle is indistinguishable from a straight line and you have the half-plane model.
 
  • #13
Interesting, huh?! =)
 

FAQ: Riemannian Geometry: What is It?

What is Riemannian Geometry?

Riemannian Geometry is a branch of mathematics that studies the geometric properties of curved surfaces and spaces. It was developed by German mathematician Bernhard Riemann in the 19th century and has applications in various fields such as physics, engineering, and computer science.

How is Riemannian Geometry different from Euclidean Geometry?

Riemannian Geometry deals with curved spaces, while Euclidean Geometry deals with flat spaces. In Riemannian Geometry, the rules for measuring distances, angles, and curvature are different from those in Euclidean Geometry. Riemannian Geometry also allows for the study of higher-dimensional spaces, while Euclidean Geometry is limited to three dimensions.

What is the significance of Riemannian Geometry?

Riemannian Geometry is essential in understanding the physical world. It is the mathematical framework for Einstein's theory of general relativity, which describes the laws of gravity and the behavior of massive objects in the universe. It is also used in other areas of physics, such as quantum mechanics and cosmology.

How is Riemannian Geometry applied in other fields?

Riemannian Geometry has a wide range of applications in various fields, including computer graphics, computer vision, and robotics. It is also used in optimization problems, such as finding the shortest path between two points on a curved surface. Additionally, Riemannian Geometry has applications in statistics and machine learning for analyzing and modeling complex data sets.

What are some famous theorems in Riemannian Geometry?

One of the most famous theorems in Riemannian Geometry is the Gauss-Bonnet Theorem, which relates the curvature of a surface to its topology. Other notable theorems include the Hopf-Rinow Theorem, which characterizes the completeness of a Riemannian manifold, and the Nash Embedding Theorem, which states that any Riemannian manifold can be isometrically embedded into a higher-dimensional Euclidean space.

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