Riemannian manifolds in physics

Riemannian manifolds are mathematical spaces used in physics to describe how distances and shapes behave in curved environments, like spacetime. Unlike flat surfaces, such as a sheet of paper, these manifolds can bend and curve, affecting how objects move and interact. For example, in general relativity, gravity is understood as the curvature of spacetime itself, and Riemannian geometry provides the rigorous framework to analyze this curvature. Essentially, Riemannian manifolds help physicists understand and calculate the effects of curvature on physical phenomena, offering a precise way to describe the universe’s complex, curved structure.