First-order method for approximating parallel transport of a vector along a curve
In the theory of general relativity, and differential geometry more generally, Schild's ladder is a first-order method for "approximating" parallel transport of a vector along a curve using only affinely parametrized geodesics. The method is named for Alfred Schild, who introduced the method during lectures at Princeton University.
Construction.
The idea is to identify a tangent vector "x" at a point formula_1 with a geodesic segment of unit length formula_2, and to construct an approximate parallelogram with approximately parallel sides formula_2 and formula_4 as an approximation of the Levi-Civita parallelogramoid; the new segment formula_4 thus corresponds to an approximately parallel translated tangent vector at formula_6
Formally, consider a curve γ through a point "A"0 in a Riemannian manifold "M", and let "x" be a tangent vector at "A"0. Then "x" can be identified with a geodesic segment "A"0"X"0 via the exponential map. This geodesic σ satisfies
formula_7 
formula_8
The steps of the Schild's ladder construction are:
Approximation.
This is a discrete approximation of the continuous process of parallel transport. If the ambient space is flat, this is exactly parallel transport, and the steps define parallelograms, which agree with the Levi-Civita parallelogramoid.
In a curved space, the error is given by holonomy around the triangle formula_10 which is equal to the integral of the curvature over the interior of the triangle, by the Ambrose-Singer theorem; this is a form of Green's theorem (integral around a curve related to integral over interior), and in the case of Levi-Civita connections on surfaces, of Gauss–Bonnet theorem.