Levi-Civita connection and parallel transport

A connection differentiates vector fields through an operation $\nabla_XY$ that is linear in the direction X over smooth functions and obeys a product rule in Y. Metric compatibility says differentiating an inner product is the sum of the two differentiated factors. Torsion freedom says $\nabla_XY-\nabla_YX=[X,Y]$. The fundamental theorem of Riemannian geometry asserts existence and uniqueness of a connection satisfying both properties for every smooth Riemannian metric.

In coordinates, write the derivatives of coordinate basis vectors using Christoffel symbols. Torsion freedom makes the lower two indices symmetric, while metric compatibility determines the displayed formula from derivatives of the metric and its inverse. These coefficients are not the components of a tensor: a coordinate change introduces extra derivative terms. Nonzero Christoffel symbols can therefore arise in a flat plane when the coordinate basis itself varies from point to point.

Along a smooth curve γ, a field V has components relative to basis vectors that may themselves change. Its covariant derivative therefore combines the ordinary component derivative with a Christoffel correction involving the curve velocity. Setting that derivative to zero is a linear ordinary differential equation. An initial vector determines a unique parallel field along the whole finite smooth curve segment, giving a linear isomorphism between the endpoint tangent spaces.

Metric compatibility makes the inner product of two parallel fields constant along the curve, so parallel transport is an isometry of tangent spaces. This does not mean the endpoint vectors have equal numerical coordinate columns: different endpoint bases may have different lengths and directions. Transport can also depend on the chosen path. Around loops, the resulting linear map is called holonomy; curvature and global topology influence it, so there is no universal path-independent identification of all tangent spaces.

Away from the origin, the Euclidean plane has polar metric $dr^2+r^2d\theta^2$. Its nonzero coordinate coefficients are $\Gamma^r_{\theta\theta}=-r$ and $\Gamma^\theta_{r\theta}=\Gamma^\theta_{\theta r}=1/r$. The radial basis has unit length, but the angular coordinate basis has length r. A field's angular coordinate component must account for this scale, in addition to the rotation of the frame. Forgetting either feature produces an incorrect transport calculation.

At a single point one may instead use the orthonormal frame consisting of the radial direction and the unit angular direction. Its component rules differ from those in the coordinate frame because the angular vector has been divided by r. Calculations should declare the chosen basis rather than mix their coefficients. Polar coordinates also fail at the origin, but the Euclidean connection remains smooth there in Cartesian coordinates; a chart singularity need not be a geometric singularity.