For a Riemannian manifold, the sectional curvature K(σ) of a two-dimensional tangent plane σ is the number obtained by evaluating the Riemann curvature tensor on an orthonormal basis of σ and normalizing; it measures the Gaussian curvature of the surface obtained by geodesics tangent to σ.

Principle

Sectional curvature extracts the curvature associated to a specific tangent 2-plane and is determined algebraically by the Riemann curvature tensor; it controls how geodesics initially tangent to the plane spread or focus.

Demonstration

On the round n-sphere of radius r every 2-plane has constant sectional curvature K = 1/r^2; geodesic triangles on the sphere have angle sums greater than π consistent with positive sectional curvature.

Misapplication

Assuming knowledge of sectional curvature in a finite set of planes suffices to determine the full curvature tensor in dimensions greater than three is incorrect; directional information may not reconstruct all components of Riemann curvature.

Consequence

Signs and bounds on sectional curvature imply strong geometric and topological results (comparison theorems, control of conjugate points, rigidity phenomena); uniform positivity or negativity has profound implications for global geometry.

Reversal

Flatness is the reversal: zero sectional curvature for every plane characterizes locally isometric-to-Euclidean manifolds, inverting the phenomena of focusing/defocusing of geodesics.

Boundary

Defined only for nondegenerate two-planes in the tangent space of a Riemannian or pseudo-Riemannian manifold; not applicable without a metric, fails for degenerate or singular metrics, and in Lorentzian signature one distinguishes timelike, spacelike, and null sectional curvatures.

Semantic Tension

There is tension between sectional curvature as a directional, fine-grained invariant and scalar or Ricci curvatures which average over directions; results that use sectional bounds are stronger but harder to verify from coarser invariants.

Synthesis

Sectional curvature is the directional Gaussian curvature attached to a tangent 2-plane, computed from the Riemann tensor; it governs local geodesic behaviour and feeds into averaged notions (Ricci, scalar) while retaining the most detailed directional curvature information.