Connections and curvature

A connection on the tangent bundle assigns a vector field ∇ₓY to two vector fields X and Y. It is linear over smooth functions in X and obeys a product rule in Y: differentiating fY produces X(f)Y plus f∇ₓY. In coordinates, derivatives of Y's components are supplemented by terms containing Christoffel coefficients. Those coefficients compensate for the changing coordinate basis. They are not tensor components, because their transformation law includes second derivatives of the coordinate change.

A connection can be chosen without a metric, and an arbitrary choice can have torsion or fail to preserve lengths. For a Riemannian metric there is exactly one connection that is both torsion-free and metric-compatible: the Levi-Civita connection. Torsion-free means ∇ₓY−∇ᵧX=[X,Y]. Metric compatibility means differentiation respects the inner product, so parallel transport preserves lengths and angles. Neither condition alone gives the uniqueness theorem. The two requirements together tie differential comparison to the declared geometry.

Use the convention $R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$. The bracket correction removes effects caused merely by using noncommuting vector fields. With this convention, sectional curvature for orthonormal vectors u,v is g(R(u,v)v,u); a unit round sphere has sectional curvature +1. Other books negate R or arrange its indices differently. A sign disagreement should therefore be resolved by comparing definitions and a known example, not by replacing symbols until formulas look alike.

Unlike the Christoffel coefficients, curvature is tensorial: its value at a point depends only on the input vectors at that point, not on how they were extended to nearby vector fields. One can choose coordinates making Levi-Civita coefficients vanish at a given point, but cannot erase a nonzero curvature tensor there by relabeling coordinates. Curvature also governs infinitesimal parallel transport around small loops. Global transport around large noncontractible loops can contain information not settled by a local curvature test alone.

The Euclidean plane away from the origin has polar metric dr²+r²dθ². Its nonzero Levi-Civita coefficients include Γʳθθ=−r and Γθrθ=Γθθr=1/r. These terms describe the rotation and scaling of the coordinate basis. They do not imply intrinsic curvature. Substituting them into the curvature formula cancels the derivative and product terms. A coordinate singularity at r=0 is likewise not a geometric singularity of the Euclidean plane; the chart simply fails there.

For an affinely parametrized straight Euclidean trajectory, the polar-coordinate geodesic equation reads r̈−rθ̇²=0 and θ̈+2ṙθ̇/r=0. A nonzero radial coordinate acceleration can therefore coexist with zero intrinsic acceleration. This is a practical diagnostic whenever an unfamiliar chart produces complicated equations. First ask whether the terms arise from changing basis vectors, then compute an invariant such as curvature. The visual bending of embedded arrows and the number of nonzero coefficients are unreliable substitutes for that invariant.