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the Parallel Transport of Vectors on a Surface |
Tullio Levi-Civita (lay-vee chee-vee-tah), the great Italian mathematician, formulated the concept of the parallel transport of vectors on a surface a long time ago but, despite its importance in mathematics and physics, there is little available in the literature to give one an intuitive grasp of the concept. This is an attempt to provide such a grasp. In the material below vectors are denoted in red.
If a vector v at a point p in a surface M is parallel-transported around a closed curve α (i.e., back to p) the resulting vector v* is not necessarily the same as v. This phenomenon is called holonomy. Parallel transport rotates all vectors through the same angle, which is called the holonomy angle of the curve α in the surface M. This angle is measured modulo 2π.
Suppose M is a sphere and α is a circle of constant latitude φ. The holonomy angle ψ_{α} is equal to -2πsin(φ). Only at the equator wwheere φ=0 does a vector return to itself under parallel transport. Near the poles where φ is close to ±π/2 the holonomy angle is close to ±2π so the difference between a vector and its parallel transport is not small.
If α is homotopically equivalent to a point via the surface S enclosed within M by α then the holonomy angle of α in M is
where K is the Gaussian curvature of M.
The Gaussian curvature of a sphere of radius r is 1/r^{2}. The area of a ribbon of width rdφ at latitude φ is (2πrcos(φ))rdφ so the value of the curvature integral over the area between latitude φ_{0} and the pole at π/2 is
The area between latitude φ_{0} and the pole is the area in the sphere enclosed by a circle of constant latitude φ_{0}. Modulo 2π, 2π[1 - sin(φ_{0}] is equal to -2πsin(φ_{0}), therefore the holonomy angle for parallel transport around a circle of constant latitude is equal to -2πsin(φ_{0}).
Erwin Kreyszig in his Differential Geometry gives two useful propositions on parallel transport.
For a disk with a unit radius the circumference is 2π. The length of the bottom edge of the cone is 2πsin(φ) so the length of the arc of the gap is 2π-2πsin(φ)=2π[1-sin(φ)].
In the diagram shown below a vector is parallel displaced along the edge of the cone. For the unrolled cone this vectors are parallel in the usual sense. When the cone is rolled up and the edges edges of the gap are brought together there is a difference in the angle and the difference is equal to the gap angle of the flattened cone.
This operation of flattening may also be applied to pyramids (without bottoms). In the case of pyramids there is no apex angle properly defined but the gap angle is easily computed. It is 2π minus the sum of the angles of the sides of the pyramid at the apex.
The second proposition from Kreyszig is that if two surfaces M_{1} and M_{2} are tangent along a curve α then vectors which are parallel to each other in M_{1} along α are also parallel to each other in M_{2} along α. This proposition can be used to find the holonomic angle for parallel transport around a sphere by considering the construction below.
There is a special problem for cones and pyramids because there is an essential singularity at the apex. The Gaussian curvature is zero everywhere in a cone or a pyramid except at the apex. Any curve in the the cone or pyramid which does not encircle the apex will have a holonomic angle of of zero, but any curve which does encircle the apex will have a holonomic angle equal to the gap angle, which for the cone is 2π[1-sin(φ)} (where φ is the apex angle of the cone) and for the pyramid 2π-Σα_{i} (where the {α_{i}} is the set angles of the sides of the pyramid at the apex).
The proposition Kreyszig for tangent surfaces can also be used to get holonomy angles for parallel tranport on a torus. Since the holonomy angles for parallel transport along circles of constant latitude on a sphere are known one needs only construct a tangency of sphere on a torus, as shown below:
Consider the four circles on the torus shown. For α_{1} the holonomic angle is zero because a tangent sphere in the interior of the torus would have α_{1} tangent on a great circle (say on its equator) and the holonomic angle for parallel transport around the equator of a sphere is zero.
For circle α_{2} which is at the top of the torus the holonomic angle can be found by parallel transport in the plane which is tangent to α_{2} and the top of the torus.
The other circles may be treated by finding the tangent circles to a cross section of the torus.
The diagrams below show what the latitude angle for the torus is compared to that of the tangent sphere. If the latitude angle is less than π/2 then the latitude for the torus is the same as that of the tangent sphere. If the latitude angle φ for the torus is greater than π/2 then the latitude angle for the tangent sphere is (π-φ). This corresponds to the inner portion of the torus. It is the same a the angle measured from the inner equator of the torus rather than from the outer equator.
Since the holonomy angle for parallel transport along a circle of constant latitude ζ is equal to -2πsin(ζ) the holonomy angle forr parallet transport along a circle of constant latitude on the inner side of a torus is -2πsin(π-φ)=-2πsin(φ).
The holonomic angle could have been equally computed using cones which are tangent to the torus.
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