Translations, Parallel Transport,
and Holonomy in the Poincare Model

Translation in Euclidean geometry is a fairly simple process. If we imagine walking along a straight line, then we are essentially translating ourselves along this line. If we now carry a stick at a fixed angle as we walk, then the end of this stick is being transported in a parallel fashion along with us as we walk. The angle that the stick makes with the line is constant as we move. We are performing a parallel transport of the direction that the stick makes with the line.

If we now imagine that we are parallel transporting the stick around the edges of a triangle ABC, starting and ending at A, then the direction that the stick makes after returning to A will be identical to the direction that the stick originally made with the path at A.

Translation in Non-Euclidean geometry is not quite so simple. Let us look at what happens as we translate a segment around a triangle. Start with triangle ABC and edge AD. Translate AD from A to B by multiple-selecting A and then B and then choosing "Vector" from the "Mark" menu in the Transform Panel. When the dialog box pops up choose "Rectangular" for a simple translation and then hit "Okay". We have now defined a hyperbolic translation from A to B. Now select segment AD and hit the now-active translate button in the Transform Panel. Segment AD will be parallel transported to point B resulting in the segment BE, as shown.

Next, define a new translation vector from point B to point C and translate segment BE, resulting in segment CF, as shown.

Finally, define a translation vector from C to A and translate CF, resulting in segment AG, as shown.

Clearly, the original segment AD and the parallel transport around the triangle, segment AG, are not in the same direction, as was the case in Euclidean Geometry. To see how much the segment has changed let's measure the angle DAG, as this measures the net rotation of AD to AG. (Note: In Euclidean Geometry, the combination of two translations is again a translation. The example we just constructed shows that the combination of translations in Hyperbolic geometry need not be a translation, and in fact is a rotation in this case.) As a comparison, let's also compute the defect of triangle ABC.

That is interesting, the defect and the net angle change appear complimentary. Let's use the Calculator to add these two measurements together. Then, we will evaluate the sum and add this new measure back to the canvas. (If you need help on using the Calculator check the help section on that topic here.)

Does this relationship persist? Let's move point A around and check...

The relationship does hold. In fact, this relationship is connected with a very important idea in geometry -- that of holonomy.

We define the Holonomy of triangle ABC to be the smallest angle measured counterclockwise from the original position of a segment AD to the final position of AD (which in this example is segment AG) as AD undergoes parallel transport counterclockwise around the triangle.

There are several conclusions we can tentatively draw from this example.

The holonomy and the defect are linearly related. In fact, if we treat an angle greater than 180 degrees as a negative angle (thus less than 180), we would have that the holonomy is equal to the negative of the defect.
Since the defect is proportional to the area of the triangle, we have that the holonomy and the negative area of the triangle are proportional. This fact is a direct consequence of a very deep formula in geometry called the Gauss-Bonnet Formula. This formula relates the area, holonomy, and Gaussian Curvature of a region on a surface. Specifically, it says that for a region on a surface with constant Gaussian Curvature (the Poincare model has Gaussian Curvature of -1) the holonomy of a region (angle change measured in radians) is equal to the Gaussian Curvature of the surface multiplied by the area of the region.