Riemannian Connection On A Surface - Parallel Transport

Parallel Transport

Given a curve in the Euclidean plane and a vector at the starting point, the vector can be transported along the curve by requiring the moving vector to remain parallel to the original one and of the same length, i.e. it should remain constant along the curve. If the curve is closed, the vector will be unchanged when the starting point is reached again. This is well known not to be possible on a general surface, the sphere being the most familiar case. In fact it is not usually possible to identify simultaneously or "parallelize" all the tangent planes of such a surface: the only parallelizable closed surfaces are those homeomorphic to a torus.

Parallel transport can always be defined along curves on a surface using only the metric on the surface. Thus tangent planes along a curve can be identified using the intrinsic geometry, even when the surface itself is not parallelizable.

Parallel transport along geodesics, the "straight lines" of the surface, is easy to define. A vector in the tangent plane is transported along a geodesic as the unique vector field with constant length and making a constant angle with the velocity vector of the geodesic.

For a general curve, its geodesic curvature measures how far the curve departs from being a geodesics; it is defined as the rate at which the curve's velocity vector rotates in the surface. In turn the geodesic curvature determines how vectors in the tangent planes along the curve should rotate during parallel transport.

A vector field v(t) along a unit speed curve c(t), with geodesic curvature k_g(t), is said to be parallel along the curve if

• it has constant length
• the angle θ(t) that it makes with the velocity vector satisfies

This yields the previous rule for parallel transport along a geodesic, because in that case k_g = 0, so the angle θ(t) should remain constant. The existence of parallel transport follows from standard existence theorems for ordinary differential equations. The above differential equation can be rewritten in terms of the covariant derivative as

This equation shows once more that parallel transport depends only on the metric structure so is an intrinsic invariant of the surface. Parallel transport can be extended immediately to piecewise C1 curves.

When M is a surface embedded in E3, this last condition can be written in terms of the projection-valued function P as

or in other words:

The velocity vector of v must be normal to the surface.

Arnold has suggested that since parallel transport on a geodesic segment is easy to describe, parallel transport on an arbitrary C1 curve could be constructed as a limit of parallel transport on an approximating family of piecewise geodesic curves.

This equation shows once more that parallel transport depends only on the metric structure so is an intrinsic invariant of the surface; it is another way of writing the ordinary differential equation involving the geodesic curvature of c. Parallel transport can be extended immediately to piecewise C1 curves.

The covariant derivative can in turn be recovered from parallel transport. In fact can be calculated at a point p, by taking a curve c through p with tangent X, using parallel transport to view the restriction of Y to c as a function in the tangent space at p and then taking the derivative.