Curvature measures the failure of covariant derivatives to commute

For the torsion-free Levi-Civita connection in a coordinate basis, the commutator of two covariant derivatives acting on a vector is algebraic in that vector. Expanding the two derivatives gives derivative-of-connection terms and quadratic connection terms. Their particular combination transforms as a tensor even though none of the connection coefficients does. The last two curvature indices are antisymmetric because they identify the two derivative directions. Reversing their order reverses the sign of the action.

Define R(X,Y)Z as ∇X∇YZ−∇Y∇XZ−∇[X,Y]Z. The vector Z occupies the second curvature slot in the component convention below, while X and Y occupy the last two. This matters in a noncoordinate frame, whose commutator may not vanish. Torsion would also modify the simple coordinate commutator formula. Throughout this module the metric signature is −+++ and the connection is torsion free and metric compatible; alternative curvature conventions in other references must be translated before comparing components.

In the flat polar plane, Γʳφφ=−r and Γφrφ=1/r. A typical curvature component contains a radial derivative −1 and a quadratic-connection contribution +1, which cancel exactly. Keeping only derivative terms would incorrectly label the plane curved. At the polar origin the coordinate formula cannot be evaluated by inserting r=0, because that point is outside this chart. Cartesian coordinates show regular flat geometry through the origin. The tensor criterion is vanishing curvature on regular patches, not small or finite values of every individual polar coefficient.

On a round two-sphere the cancellation instead leaves positive curvature in our convention, with orthonormal sectional curvature 1/a² for radius a. Parallel transport around a sufficiently small loop then depends on its oriented area. The direction of a loop matters, and reversing it reverses the leading holonomy. Infinitesimal holonomy is local curvature information. Noncontractible loops can additionally detect global identifications, so flatness alone need not establish path independence around every loop of an entire spacetime.

A freely falling observer can set local connection coefficients to zero at one event, eliminating the coordinate appearance of a gravitational acceleration there. Neighboring freely falling trajectories can still converge or separate. Geodesic deviation measures that relative effect to first order in their separation. Positive Rⁱ₀ⱼ₀ in a chosen spatial eigen-direction produces focusing under the displayed sign convention. The curvature must be evaluated along the reference trajectory, and treating it as constant is an additional short-time or special-geometry approximation.

The oscillator lab illustrates one constant focusing component after normalization. Reset and set both controls to damping ζ=0 and frequency ω₀=1; after one Step to t=0.5 s predict x=cos0.5≈0.878. Reset and set ζ=0 and ω₀=2; one Step predicts cos1≈0.540. Both start with displacement one and velocity zero. This matches the mathematical form of a constant positive tidal eigenvalue after assigning suitable units. It contains no spacetime metric, tensor components, or changing curvature, and should not be treated as a general geodesic-deviation solver.