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2

Short answer: yes. But what do the original "straight lines" mean now? they cannot be defined in a nice way in the new metric, because the natural geometric entities now are geodesics, and a quadrilateral of four right angles does not exist (apart from possible special choice of corners). You must define your volume in a correct way (see below) Long answer: ...

0

You are on the right track, but it is a bit more complicated than that. At every point in space around a magnet there is a magnetic field. This field isn't just $+$ or $-$'s, instead there is an arrow (a vector, in mathematical language) pointing in some direction, as in this picture: ...

1

You ask the question of whether the title for the Wikipedia article should be "Curved Spacetime" instead of "Curved Space". The answer is a resounding no, leave it as is. The article itself covers strictly the mathematics of any curved space and is not specific to physics contexts. As is, the usage of "Space" does not mean purely spatial and not temporal ...

3

Your argument is incorrect: The curvature of a spatial slice is coordinate-independent. What is true is that in general relativity, there is a priori no preferred spatial slicing. For example, de Sitter spacetime (a universe dominated by cosmological constant) can be sliced into positively curved, negatively curved or flat spaces. When we say that our ...

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