**Holonomy and curvature**
Holonomy is from the Greek

*holos* meaning "whole" and the root

*nom* (this is according to Wiktionary), and the latter is related to the notion of "naming or numbering".

Also to geodesy and surveying. Romans built straight roads for absolute bloody miles, using a fairly simply constructed device, the

*gnomon*.

This employed, or deployed plumb weights and a revolving upper arm to first align a central pole upright, then sight along two plumb lines to a distant

*observer*.

You can read all about holonomy

here, and this is a nice pretty picture:

I tried to do a sort of explanation:

Transport around a sphere as in the above diagram, is the classical example of anholonomy introduced by the curvature of the sphere, when the transported object has an initial direction, which in the case of being directed towards a pole of the sphere from the equator (any point lying on the diameter), then back to a different point changes the direction.

If the object is a vector with magnitude as well as direction, the transport to a pole and back transforms the vector (an operation on the vector space), the change in direction depends on the path over the sphere alone.

Thus there are three

*sections* in such a path to the pole, from the pole, and back to the start point (x). Step 1 (T1) transports the vector over a geodesic curve = 1/4 the [oops, circumference projected to the] diameter so that it is parallel to a line on the equatorial plane, step 2 returns it to the equator along an arbitrary direction, along another 1/4 [circumference projected to a] diameter (the vector is now an outer derivative of the intersection of two planes through the pole that each bisect the sphere like the equatorial ), the remaining step that returns it to x_initial depends only on the angle at the pole between T1 and T2, so that T2 -> T0 is an arc subtended by the polar angle between T1,T2 a spherical triangle.

The

*ascent* T1 (pitch) is parallel to the geodesic curve, the

*descent* T2 (roll) is orthogonal to the angle subtended, and the final anholonomy is a result of the curvature of each connected step Tn plus "transport" = pitch, roll, yaw.

In other words a vector pointing in an arbitrary direction at the pole (with an infinite circle of directions) parallel to the equatorial plane of the sphere, which is transported to the equator along any geodesic other than parallel to the diameter it is pointing along at T0 at the pole, will 'preserve' the angular difference, when it returns to the pole.