Sectional Curvature

Differential Geometry · Axiom Academy

Understanding how Riemannian manifolds curve in two-dimensional planes 1. The Definition of Sectional Curvature Given a Riemannian manifold (M, g) and a 2-plane σ ⊂ T_pM spanned by linearly independent vectors X, Y, the sectional curvature K(σ) is defined as: where R is the Riemann curvature tensor. This formula is independent of the choice of basis X, Y for the plane σ. 2. Geometric Interpretation: Gaussian Curvature of 2D Slices The sectional curvature K(σ) has a beautiful geometric interpretation: it measures the Gaussian curvature of a 2-dimensional surface obtained by taking all geodesics through p tangent to the plane σ. This connects the intrinsic curvature of higher-dimensional manifolds to the familiar Gaussian curvature of surfaces. For a 2-dimensional surface itself, there is only one sectional curvature at each point, which is precisely the Gaussian curvature. 3. Sectional Curvature Determines the Riemann Tensor A remarkable fact is that knowing all sectional curvatures K(σ) for all 2-planes σ at each point completely determines the Riemann curvature tensor R. This is proven using the polarization identity . By expanding sectional curvatures of planes spanned by X+Z and Y+W, then X+W and Y+Z, and combining these expressions, we can isolate R(X,Y,Z,W). 4. Constant Sectional Curvature Spaces Spaces where K(σ) = c for all 2-planes σ at all points are called spaces of constant curvature . These are the most symmetric Riemannian manifolds and fall into three classes:

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