Didn't get much farther than Calc 1; is this a fancy way of saying "Earth is flat in ~~three~~ four dimensions"?
It's been a while so bits of this might be wrong. Manifolds give me PTSD.
A euclidean space is one where pythagoras' theorem is true, like on a piece of paper. Even if you roll up the paper, everything still works the same. The angles of a triangle still add to 180 etc
But the surface of a sphere doesn't work like that. You can't fold up the paper into a sphere, it's fundamentally different. Geometry changes a bit on the surface too, and you have to modify pythagoras theorem for it to describe triangles in one of these spaces. And the internal angles don't add to 180 anymore.
These are non-euclidean spaces, aka Riemannian spaces.
So the meme is saying "ha ha you're in a non-Euclidean space, it just looks like that because you're small, dumb dumb! Why don't you draw a big triangle, measure the angles and then cry yourself to sleep?"
A Riemannian manifold isn't necessarily non-Euclidean, it's just a smooth manifold with a Riemannian metric, which is just sort of a way of defining local geometry in a coherent way. Namely it's a smooth family of inner products on the tangent spaces at each point, where an inner product on the tangent space is sort of a way of comparing any two directions at a point and the smoothly varying part means that for sufficiently close points, the comparison function on their respective tangent spaces is similar.
Anyway, like "manifold" is a formalism intended to capture the idea of a "shape or space," a "Riemannian manifold" is just "a shape or space we can do geometry on."
Ah thanks! This is why I have manifold PTSD but also that was genuinely helpful.
This was covered fully in the only course I failed in my physics masters, lol.
And I just want to add as a physicist: The most interesting manifolds are the differentiable ones, because there you can do general relativity! But manifolds are also relevant in other, more unexpected places, like a pendulum: It moves on a submanifold of R^3 due to the constraint of the string.
soooo.....
no?
(lol)
Three Body Problem Spoiler:
"Hold my vector foil"
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