Metric and existence of parallel lines

[ShayanJ]

I don't know very much about differential geometry but from the things I know I think that the metric is somehow the quantity which specifies what kind of a geometry we're talking about(Though not sure about this because different coordinate systems on the same manifold can lead to different metrics!). So I think it should be possible to know that a particular geometry is Euclidean,Hyperbolic or elliptic by taking a look at its metric.Is it right? if yes,how is that?
Thanks

 

[maajdl]

The fact that a manifold can be represented by different coordinates doesn't mean that the manifold has no geometry.
In the same way the metric has different representations for different systems of coordinates, but it still represents the same thing / geometry.
Indeed the metric is all that is needed to describe the geometry, if it is Euclidean for example.
The way to check if -for example- it is Euclidean, doesn't depend on the system of coordinates which is used.
For example, the total curvature will always be the same independently of the coordinate system used to calculate it.

 

[ShayanJ]

So...you mean we can say whether a geometry allows parallel lines by just looking at its metric? How?

 

[maajdl]

I was mainly reacting about your remark:

"Though not sure about this because different coordinate systems on the same manifold can lead to different metrics!"

I was meaning that the information about the geometry that is contained in the metric tensor is independent of the coordinates used. (that's the meaning of the metric being a tensor)

Regarding the notion of parallelism, you need first to define what a line is, since parallelism, as far as I know, is related to "lines". In the context of Riemannian geometry (ie based on a metric), parallelism is defined on the basis of geodesics. The existence of parallel geodesics can be verified by exploring the metric tensor.

If the distinction between Euclidean, Hyperbolic of Elliptic geometry is defined by comparing the sum of the angles of a triangle to Pi, then the metric tensor is definitively able to detect this distinction. This is because, 1) the metric tensor is all that you need to define parallel transport and "lines" and 2) it defines the geometry in the tangent space (distances and angles).

 

[blue_raver22]

for your question! Your understanding of the role of the metric in geometry is correct. The metric is a mathematical concept that defines the distance between points in a space. In differential geometry, the metric is used to define the curvature of a space, which determines its geometry.

In terms of parallel lines, the metric plays a crucial role. In Euclidean geometry, parallel lines are defined as lines that never intersect and have the same slope. This can be represented mathematically using the Euclidean metric, which measures distances in a straight line. In hyperbolic geometry, the metric is different, and parallel lines are defined as lines that never intersect but have a different slope. This leads to the famous result that in hyperbolic geometry, the sum of the angles in a triangle is less than 180 degrees.

In conclusion, the metric is indeed the key to understanding the geometry of a space. By examining the metric, we can determine whether a space is Euclidean, hyperbolic, or elliptic. This has important implications for many fields of science, including cosmology, where the curvature of space is a crucial factor in understanding the universe.