Levi-Civita Parallelogramoid - Construction

Construction

A parallelogram in Euclidean geometry can be constructed as follows:

  • Start with a straight line segment AB and another straight line segment AA′.
  • Slide the segment AA′ along AB to the endpoint B, keeping the angle with AB constant, and remaining in the same plane as the points A, A′, and B.
  • Label the endpoint of the resulting segment B′ so that the segment is BB′.
  • Draw a straight line AB′.

In a curved space, such as a Riemannian manifold or more generally any manifold equipped with an affine connection, the notion of "straight line" generalizes to that of a geodesic. In a suitable neighborhood (such as a ball in a normal coordinate system), any two points can be joined by a geodesic. The idea of sliding the one straight line along the other gives way to the more general notion of parallel transport. Thus, assuming either that the manifold is complete, or that the construction is taking place in a suitable neighborhood, the steps to producing a Levi-Civita parallelogram are:

  • Start with a geodesic AB and another geodesic AA′. These geodesics are assumed to be parameterized by their arclength in the case of a Riemannian manifold, or to carry a choice of affine parameter in the general case of an affine connection.
  • "Slide" (parallel transport) the tangent vector of AA′ from A to B.
  • The resulting tangent vector at B generates a geodesic via the exponential map. Label the endpoint of this geodesic by B′, and the geodesic itself BB′.
  • Connect the points A′ and B′ by the geodesic AB′.

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