Telling if geometry is curved without <PERSON> tensor
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 <PERSON> 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:
1. If you always end up in the same point, is this enough to be able to say that the space is flat?
2. 651
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.
EDIT
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:
1. is not valid for a geometry with more than two dimensions
2. 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. 651