Differential Geometry (11): Curvature in Riemannian Geometry — Riemann, Ricci, and Scalar

Sun, 21 Nov 2021 09:00:00 +0000

Curvature is the central concept of Riemannian geometry. Intuitively, it measures how much a space deviates from being flat — how parallel lines converge or diverge, how triangles have angle excess or deficit, how volumes grow differently from Euclidean expectations. In the previous article we saw that the path-dependence of parallel transport signals the presence of curvature: a vector carried around a closed loop on $$S^2$$ returns rotated, while on $\mathbb{R}^n$ it does not. The next step is to make this precise, to extract a tensor that exactly captures this rotation, and to understand its various contractions.

Ricci on Chen Kai Blog

Differential Geometry (11): Curvature in Riemannian Geometry — Riemann, Ricci, and Scalar