Jason Shelton

Parallelism and Holonomy

When studying geometry in two-dimensional Euclidean Space, one topic that is often discussed is parallelism. Parallelism for lines is simply defined by the lines never crossing; bearing in mind that the lines are considered straight intrinsically and in Euclidean space, also extrinsically straight. Two new geometries were constructed that change the parallel postulate. The first, hyperbolic geometry, defied the parallel postulate by having infinitely many lines parallel to a specific line through a specific point. In Euclidean, there would only be one line that was parallel. In spherical or elliptic geometry, there are no parallel lines at all. This study will focus on holonomy; looking at parallelism in a new light in regards to surfaces.

To understand holonomy, we first must understand vectors. By definition, vectors are always extrinsically straight. A vector has two components, direction and magnitude; both defined from a specific location in space or off of a surface. On Euclidean two-dimensions, vectors appear to look the same as rays. In three-dimensions they are extrinsically straight, regardless of the surface or the geometry being dealt with. The most common vector used is the tangent vector. These vectors describe a pathway along a surface at a specific point on that pathway. An interpretation of this is the current direction of travel and the current velocity. Other interpretations can exist as well for magnitude, but direction of the tangent vector will always indicate direction of the pathway. When studying holonomy, not just vectors are looked at on a surface, but a parallel vector field. When a vector field is wrapped to a surface, it may not be apparent whether they are parallel. To get an idea of how a vector field is laid out on a surface, flatten the surface, or at least the section with the field, and look at the field. A parallel vector field will have all vectors pointing the same direction and have the same magnitude. When studying holonomy, the magnitude is usually not important and is left as unit length. Holonomy looks at a parallel vector field that is laid out along a curve on a surface. Each vector has an angle from the tangent on the curve that stays consistent along the entire path. The vectors can be tangent vectors as they would have a holonomic angle of 0 degrees and only if the are considered parallel when flattened into two-space. The angle that measures difference between the direction of the initial vector and the vector that comes back around the path to meet it is the holonomy of the surface along that curve. The curve of holonomy need not be a line on that surface, but it must meet itself. To ensure a curve like this, a plane must intersect the surface to create the curve.

Holonomy is more easily seen with an example of a cone. When flattened out, the cone forms a part of a circle (assuming cone angle is less than 360 degrees and greater than 0 degrees. For the cone, the curve of interest will be the outside edge which is just the circular path. In this flattened out state, a vector field is drawn starting from one point and drawn in the same direction with same magnitude. Bear in mind that these vectors start off in the same plane as the flattened cone. As the cone is folded into three-space, the vectors are distorted. This distortion will cause the vectors that come back to meet your initial vector have a different direction. For cones, the holonomy, this distortion, will always be the same as the cone angle. For other surfaces, the holonomy will vary. Spheres, another surface where holonomy is more easily studied, will have a uniform holonomy of 180 degrees. Another way to visualize the holonomy or construct is by using one initial vector on the curve of the surface. This vector will have a set angle from the surface. The vector is moved along the curve keeping its holonomic angle the same. The vector will come back around to meet the first and then the holonomy can be measured. This type of motion is termed a parallel transport. In two-space, a parallel transport for a vector is simply a translation. Because this is an isometry, it will not alter the holonomy or the general geometry of the surface as the vector moves.

Visualizing the holonomy is one thing, actually computing it is another. The computations, while easier understood are much more difficult to carry out. The initial vector is defined on the surface at a point on the curve. Usually denoted as V and set equal to two unit tangent vectors that are multiplied by trig. functions to each of their respective directions. The first vector is a tangent vector to the surface; the second is tangent to the surface but points in the way the holonomic vector makes an angle from the surface. To figure out what the vector does along the curve as a parallel transport is made, a covariant derivative must be done while setting it equal to zero. It is set equal to zero because the initial vector is first and occurs at time equal to zero. A covariant derivative is similar to a derivative as it describes slope/direction but this deals with vectors changing in directions as opposed to a curve itself. Through computations (which I will not elaborate more on) a determination that the derivative of theta (angle of difference from initial) in respect to time is equal to the negative of the metric on the parameterization of the curve. This will mean that an integral of the parameterized curve and negated will yield the holonomy itself. This makes sense because the integral will measure the distortion in the curve. The distortion will be the opposite of the holonomy so it must be negated.

Holonomy is a difficult topic for scholars to follow at times. The applications are generally very specific and hard to find as well. However, applications to exist; especially in regards to travel through earth atmosphere to space and communication with cell towers and even satellites. These computations are still altered due to other factors but scientists can use holonomy to determine how to work with the geometry of the earth.