eli5:locally euclidean vs euclidean what is the difference in layman’s terms?
In: Mathematics
You may know that there is no 2d map of the Earth that preserves all the geometry — the Mercator projection, for example, ends up enlarging places like Greenland compared to Africa. To solve this, we often use atlases, a big book of local charts. Each chart is of a much smaller area, so the distortion is not as great. Also, because each chart doesn’t “wrap around” like the globe does, we can distort it to a 2d projection without doing any cutting. This is what we mean by “locally Euclidean” — the globe is not Euclidean (flat space) but each local area (chart) does look Euclidean.
In fact this is essentially the definition of a manifold in mathematics. The globe is a 2d manifold because each chart looks like 2d Euclidean space. Spacetime is a 4d manifold because, even though we have local curvature, each chart looks like 4d Euclidean space (3 space dimensions + 1 time).
The definition of “looks like” Euclidean space has a precise mathematical definition, but roughly speaking it is some invertible transformation that doesn’t involve any cutting.
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