Sectional curvature - Knowlpedia

Let $(M,g)$ be a Riemannian manifold and let $P=\operatorname{span}(u,v)\subset T_pM$ be a two-dimensional plane in the tangent space at $p$ . With the Riemann curvature tensor convention

R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z,

the sectional curvature of $P$ is

K_g(P)=\frac{g(R(u,v)v,u)}{g(u,u)g(v,v)-g(u,v)^2}.

Basis independence

The denominator is the squared area of the parallelogram spanned by $u,v$ , so the quotient is independent of the chosen basis of $P$ .

Geometric meaning

Sectional curvature is the intrinsic curvature of the geodesic two-plane determined by $P$ , measured to second order by the metric. A metric has constant sectional curvature $k$ when $K_g(P)=k$ for every point and every tangent two-plane. In dimension two, sectional curvature is the Gaussian curvature.

Relation to Ricci curvature

For a unit vector $u$ and an orthonormal basis $u,e_2,\ldots,e_n$ of $T_pM$ ,

\operatorname{Ric}(u,u)=\sum_{j=2}^n K_g(\operatorname{span}(u,e_j)).

Thus Ricci curvature records the sum of sectional curvatures of planes containing a direction, while sectional curvature retains the individual two-plane information.

Sign convention

Some authors define the Riemann tensor with the opposite overall sign, which reverses every sectional curvature. The displayed formula and the linked Riemann curvature tensor convention fix the sign used here.

References

John M. Lee, Introduction to Riemannian Manifolds, 2nd ed., Springer, 2018. Publisher record. Relevant: Chapter 7, sectional curvature.
Manfredo P. do Carmo, Riemannian Geometry, Birkhäuser, 1992. Publisher record. Relevant: Chapter 3, sectional curvature.