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I'm trying to learn some geometry for general relativity, and I am having a bit of trouble understanding how to tell flat and curved spaces apart. Specifically, I heard that a space is flat if you can "flatten" the metric by finding a coordinate system where [tex]ds^2 = dx^2 + dy^2 + ...[/tex]. Unfortunately, if this is true, then I can show that the surface of a sphere is flat.

Let [tex]\theta[/tex] be the latitude and [tex]\phi[/tex] be the longitude on a unit sphere. Then the metric is

[tex]ds^2 = d\theta^2 + \sin^2(\theta)d\phi^2[/tex]

Now let coordinates x and y be

[tex]x = \theta[/tex]

[tex]y = \phi \sin(\theta)[/tex]

Then we get

[tex]d\theta = dx[/tex]

[tex]d\phi = \frac{dy}{\sin(\theta)}[/tex]

Plugging into the metric,

[tex]ds^2 = d\theta^2 + \sin^2(\theta)d\phi^2 = dx^2 + dy^2\frac{\sin^2(\theta)}{\sin^2(\theta)} = dx^2 + dy^2[/tex]

This is in the "flat" form.

I'm sure I've done something wrong. Maybe putting the metric in this form does not actually imply that the space is flat, or maybe I did something bad with the coordinate change?

Thanks