##  [Sectional Curvature](/sectional-curvature-0) 

 Definition

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

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

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

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

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

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

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

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

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.