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.”
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.
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.”
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.