Levi-Civita parallelogramoid

Levi-Civita's parallelogramoid

In the mathematical field of differential geometry, the Levi-Civita parallelogramoid is a quadrilateral in a curved space whose construction generalizes that of a parallelogram in the Euclidean plane. It is named for its discoverer, Tullio Levi-Civita. Like a parallelogram, two opposite sides AA and BB of a parallelogramoid are parallel (via parallel transport along side AB) and the same length as each other, but the fourth side AB will not in general be parallel to or the same length as the side AB, although it will be straight (a geodesic).

Construction

A parallelogram in Euclidean geometry can be constructed as follows:

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:

Quantifying the difference from a parallelogram

The length of this last geodesic constructed connecting the remaining points AB may in general be different than the length of the base AB. This difference is measured by the Riemann curvature tensor. To state the relationship precisely, let AA be the exponential of a tangent vector X at A, and AB the exponential of a tangent vector Y at A. Then

|A'B'|^2 = |AB|^2 + \frac{8}{3}\langle R(X,Y)X,Y\rangle + \text{higher order terms}

where terms of higher order in the length of the sides of the parallelogram have been suppressed.

Discrete approximation

Two rungs of Schild's ladder. The segments A1X1 and A2X2 are an approximation to first order of the parallel transport of A0X0 along the curve.

Parallel transport can be discretely approximated by Schild's ladder, which approximates Levi-Civita parallelogramoids by approximate parallelograms.

References

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