Riemann Curvature: Exploring 3D Curves

In summary, Riemann developed a method to determine if a surface is curved or flat without leaving the surface. This method involves using measurements taken on the surface itself to calculate the Riemann curvature tensor, which gives information about the curvature at a specific point. This concept extends to three-dimensional space and is based on the idea of tensors, which are independent of coordinate systems. However, a more detailed understanding of this concept may require a deeper understanding of mathematics.
  • #1
quietrain
655
2
Riemann Curvature?

i was watching this documentary that mentioned that riemann came up with a method to deduce whether we were on a curved surface, or on a flat surface, without leaving the surface to make the deduction.

for example, for a curved 2d surface, we know it is as such as we can see it from the 3d space we are in

but for a 3d curve, we need to be in the 4th dimension which is not possible. so he came up with a method to deduce this.
[* by the way, what is a 3d curve?]

may i know what method is this? is there anywhere i could read it on? (preferably on the internet)

thanks!
 
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  • #2


quietrain said:
i was watching this documentary that mentioned that riemann came up with a method to deduce whether we were on a curved surface, or on a flat surface, without leaving the surface to make the deduction.

for example, for a curved 2d surface, we know it is as such as we can see it from the 3d space we are in

but for a 3d curve, we need to be in the 4th dimension which is not possible. so he came up with a method to deduce this.
[* by the way, what is a 3d curve?]

may i know what method is this? is there anywhere i could read it on? (preferably on the internet)

thanks!

Hey quietrain.

I don't have the answer (sorry!), but if someone does I too am interested in this, if anyone has any insight into this problem.
 
  • #3


First, a three d curve is just a curve in three dimensional space. For example, the set of points (x, y, z) such that x= f(t), y= g(t), z= h(t), for functions, f, g, and h, form a curve in three dimensions. If you like, you can think of the parameter, t, as representing time and think of this as the path of something moving through space.

To see what Riemann was thinking, imagine a two dimensional analog. Imagine that you are a two dimensional being living on the surface of a large sphere. Locally, if the sphere where large enough (or your eyesight short enough!) you might think your space was flat. But if you were to move straight ahead, not changing direction, you would go around a circumference of the sphere and eventually get back to where you started. You would know then that your space was NOT flat since that, of course, is impossible on a flat surface. If you were to measure that distance as you went, you would know the circumference and so could calculate the radius of your sphere and then get its curvature, 1 over the radius.

At any point on a surface (or in three dimensional space) there exist a 'geodesic' (path of minimal length between two points- a straight line in flat space). We can calculate the curvature of such a path. There always exist a path where that curvature is maximum and a path where that curvature in minimum over all path through that point. Their product is the "Gaussian curvature" of the surface or space.

In order to go beyond that, Riemann used "tensors" which are an extension of the idea of vectors. The basic idea is that tensors are independent of the coordinate system- while there individual components will depend on the specific coordinate system used, if A= B, where A and B are tensors, in one coordinate system, then A= B in any coordinate system. Riemann then showed that we could use measurements in the surface itself (like our two-dimensional self measuring the distance he went) to form the "Riemann curvature tensor" that gives information about the curvature- and that generalizes easily to three dimensions. The "Riemann curvature" at a point, specifically, is the "contraction" of that tensor.

If all you know about "curvature" is a reference on a documentary then a more detalied explanation is probably beyond your mathematics- and perhaps beyond mine- it has been a long time since I took differential geometry.
 
  • #4


wow... thanks!
 
  • #5


I am familiar with Riemann curvature and its significance in the field of mathematics and physics. Riemann curvature is a measure of how curved a space is at any given point. It was first introduced by mathematician Bernhard Riemann in the 19th century and has since been used in various fields, including general relativity and differential geometry.

In regards to your question about 3D curves, this refers to a curve that exists in three-dimensional space. This could be a physical object or a mathematical representation.

As for Riemann's method to deduce whether we are on a curved or flat surface, this is known as the Riemann curvature tensor. It is a mathematical tool that allows us to calculate the curvature of a space at a specific point. This method has been extensively studied and applied in various fields, and you can find more information about it in textbooks or online resources on differential geometry and general relativity.
 

1. What is Riemann Curvature and why is it important?

Riemann Curvature is a mathematical concept used in differential geometry to measure the curvature of a space or curve. It is important because it allows us to understand the shape of 3D curves and surfaces, which has applications in fields such as physics, engineering, and computer graphics.

2. How is Riemann Curvature calculated?

Riemann Curvature is calculated using a mathematical formula involving the second derivatives of a curve or surface. This formula takes into account how the curve or surface curves in different directions, giving us a value for the curvature at any given point.

3. Can Riemann Curvature be negative?

Yes, Riemann Curvature can be negative. This indicates that the curve or surface is curving in a concave manner. Conversely, a positive Riemann Curvature indicates a convex curvature, and a value of zero indicates a flat curve or surface.

4. How does Riemann Curvature relate to Gaussian curvature?

Riemann Curvature is a component of Gaussian curvature, which is a measure of the intrinsic curvature of a surface. Gaussian curvature takes into account the curvature in all directions, while Riemann Curvature only measures the curvature in a specific direction.

5. What are some real-world applications of Riemann Curvature?

Riemann Curvature is used in a variety of real-world applications, such as studying the curvature of the Earth's surface, designing curved structures in architecture and engineering, and creating realistic computer-generated graphics. It also has applications in physics, particularly in understanding the curvature of spacetime in Einstein's theory of general relativity.

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