# Metric and existence of parallel lines!

Gold Member
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

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maajdl
Gold Member
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.

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

maajdl
Gold Member