Riemann curvature and sectional curvature

Use $R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$. This expression is linear over smooth functions in each of X, Y, and Z, so its value at a point depends only on the three tangent vectors there. The bracket correction is essential to that tensoriality. With this convention, a constant-curvature space satisfies $R(X,Y)Z=K(g(Y,Z)X-g(X,Z)Y)$, and the standard round sphere has positive K.

The tensor is antisymmetric in its first two inputs. Metric compatibility supplies additional symmetries after pairing its output with a fourth vector, and torsion freedom gives the first Bianchi identity. These restrict the independent components dramatically, but they do not make the raw Christoffel symbols tensorial. When consulting a source with the opposite convention, reverse its curvature operator and adjust all related formulas together; changing only the word positive would corrupt the Jacobi equation and comparison inequalities later.

The denominator in the sectional quotient is the squared area of the parallelogram spanned by u and v. Tensor symmetries show that changing the spanning basis multiplies numerator and denominator by the same squared determinant, leaving their ratio unchanged. For an orthonormal pair, the denominator is one. Sectional curvature therefore measures the selected two-dimensional tangent direction, not the lengths of the chosen coordinate basis vectors. In dimension two there is only one tangent two-plane, giving Gaussian curvature.

On a round sphere of radius ρ, every tangent two-plane has curvature $1/\rho^2$. More generally, multiplying a metric by c² leaves its Levi-Civita connection and curvature operator of type (1,3) unchanged, because the scaling is constant and cancels in the Christoffel formula. Pairing the curvature output gains c², while the area denominator gains c⁴, so sectional curvature is divided by c². Keeping track of tensor type prevents an apparent contradiction between an unchanged connection and changed numerical curvature.

In the Euclidean polar chart, coordinate fields commute and the connection coefficients from the preceding lesson are nonzero. Computing curvature requires differentiating those coefficients and combining the connection products. The terms cancel exactly. This cancellation reflects the underlying Euclidean geometry, even though the chart basis rotates and stretches. A singular expression at the polar origin is a failure of the chart, not evidence that the plane has a geometric curvature singularity.

Curvature is local, while some properties of geodesics depend on global topology. A flat cylinder and a flat plane both have zero sectional curvature but differ in their closed routes and global exponential maps. Conversely, a surface can be bent in a surrounding space without changing its intrinsic metric, as rolling a flat sheet into a cylinder illustrates locally. Intrinsic sectional curvature should therefore be distinguished from the ambient bending of an embedded curve or surface.