The sectional curvature K(σ) of a 2-plane σ = span{X, Y} ⊂ TpM is defined as K(σ) = g(R(X,Y)Y, X) / (g(X,X)g(Y,Y) - g(X,Y)²). The denominator is the squared area of the parallelogram spanned by X and Y. Why is this normalization needed?
ATo make K(σ) independent of the choice of basis {X, Y} for σ
BTo ensure K(σ) is always positive
CTo make K(σ) equal to the scalar curvature
DTo cancel the metric dependence so K(σ) is a topological invariant
The numerator g(R(X,Y)Y, X) depends on the specific vectors X, Y chosen to span σ — rescaling X by λ rescales the numerator by λ². Dividing by the squared area of the parallelogram (which scales the same way) makes the quotient depend only on the 2-plane σ, not on the basis. K(σ) is NOT always positive (hyperbolic space has K < 0), NOT equal to scalar curvature (that is a different contraction), and NOT a topological invariant (it depends on the metric).
Question 2 True / False
If a Riemannian manifold has constant sectional curvature K at every point and for every 2-plane, then it is locally isometric to one of three model spaces: Euclidean space (K=0), a sphere of radius 1/√K (K>0), or hyperbolic space of curvature K (K<0).
TTrue
FFalse
Answer: True
This is a fundamental classification theorem. Constant sectional curvature completely determines the local geometry — the Riemann tensor takes the form Rᵢⱼₖₗ = K(gᵢₖgⱼₗ - gᵢₗgⱼₖ), and the metric is locally that of the unique model space with that curvature. Complete, simply connected manifolds of constant curvature are exactly the three space forms: ℝⁿ, Sⁿ, Hⁿ. Non-simply-connected space forms are quotients of these by discrete isometry groups (e.g., ℝPⁿ = Sⁿ/ℤ₂, flat torus = ℝⁿ/ℤⁿ).
Question 3 Short Answer
In dimension 2, the sectional curvature at a point is the same as the Gaussian curvature. In higher dimensions, the sectional curvature contains strictly more information than the Ricci or scalar curvature. Why?
Think about your answer, then reveal below.
Model answer: In dimension 2, there is only one 2-plane at each point (the entire tangent plane), so sectional curvature is a single number — which equals the Gaussian curvature. In higher dimensions, there are infinitely many 2-planes at each point, and the sectional curvature varies over them. The Ricci curvature in a direction v averages sectional curvatures of planes containing v, and the scalar curvature averages further. These averages lose information: manifolds with the same Ricci curvature can have different sectional curvatures. The sectional curvature function determines the full Riemann tensor, while Ricci and scalar curvature do not (in dimension ≥ 4).
The fact that sectional curvature determines the full Riemann tensor is a consequence of the symmetries of R: knowing g(R(X,Y)Y,X) for all X,Y determines R by polarization. This is analogous to how a symmetric bilinear form is determined by its associated quadratic form.