In his book "Riemannian geometry" Do Carmo said
The curvature measures the amount that a riemannian manifold deviates from being euclidean
My question is
How does the curvature measure this amount? In other words, why the value and sign of curvature describe the topography of the manifold?
I think the best website I found that describes it rigorously is http://www.kjm.pmf.kg.ac.rs/pub/12617190662534_13.pdf. In that website it simplifies to taking an associated curve in the euclidean plane and seeing how much 'inverse distance'it deviates from a curve, that's why derivatives are used extensively.