If you're only interested in telling whether a certain geometry is flat or curved, and you do not need to know in which way it is curved, do you still need the Riemann tensor?
When I try to visualize something flat, I think of a sheet of paper, and for something curved I think about the surface of a sphere. On the sheet of paper, if you first go a specific distance A to the right, then turn 90 degrees to the left and go a disance B forward again, you'll end in the same point as when you first start going up until you've traveled a distance B, turn 90 degrees to the right and go a distance A forward. If you do the same thing on the surface of the sphere, you won't end up in the same point anymore (I hope the image clarifies my reasoning).
The things I was wondering about:
- If you always end up in the same point, is this enough to be able to say that the space is flat?
- Is this kind of reasoning sufficient to decide if a certain geometry is curved or not? Or do you need the Riemann curvature tensor for this?
I started thinking about this in the context of special and general relativity, but I think this 2D example captures the essence of my question.
Having slept on this for a bit, I think that this reasoning may be valid in a 2D geometry, but no longer for even just a 3D space: in a 3D environment, I could imagine two 'people' performing a similar experiment. If they don't end up in the same spot, I think it's fair to deduce that the space is curved. However, if they do end up in the same spot, I do not believe that one can conclude that space is flat: I imagine it's possible that the two inhabitants of the 3D world could find that they are rotated with respect to each other.
So my guess would be that:
- is not valid for a geometry with more than two dimensions
- the specified reasoning is not sufficient in general, and the Riemann tensor is really necessary
It would be great if someone could confirm or correct this.