What is Riemann's method for determining curvature in 3D spaces?

[quietrain]

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!

 

[chiro]

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.

 

[HallsofIvy]

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.

 

[quietrain]

wow... thanks!