Covariant Derivative & Parallel Transport

Differentiate vector fields on a surface; transport them along curves.

The idea

The covariant derivative differentiates a tangent vector field along a curve on a surface, giving an answer that stays tangent to the surface.

Let $V(t)$ be a field of tangent vectors along a curve on a surface $S$. The ordinary derivative $\frac{dV}{dt}$ is a vector in $\mathbb{R}^{3}$ that in general points off the surface. Split it into a component tangent to $S$ and a component along the normal, and keep the tangential component: that is the covariant derivative $\frac{DV}{dt}$. The discarded normal component is extrinsic — it records how the surface bends in space — while $\frac{DV}{dt}$ is intrinsic: it can be computed from the first fundamental form alone, so bending the surface without stretching it changes no covariant derivative.

A field is parallel along the curve when $\frac{DV}{dt} = 0,$ and carrying a vector along a curve so that it stays parallel is parallel transport. Transport preserves the lengths of vectors and the angles between them. It does not preserve independence of the path: on a curved surface a vector transported around a closed loop can return rotated. That rotation angle, the holonomy of the loop, equals the total Gaussian curvature enclosed by the loop, and in the plane it is always zero.

Ways to work on it

• Walkthrough. Covariant derivative as the tangential part, parallel fields, and geodesics.
• Practice. Holonomy of transport around a geodesic triangle.
• Hardest. Parallel transport around a latitude circle and its curvature limit.

Not sure where to start? Take the ten-question placement test.