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 sidesAA′ and BB′ of a parallelogramoid are parallel and the same length as each other, but the fourth side A′B′ will not in general be parallel to or the same length as the side AB, although it will be straight.
Slide the segment AA′ along AB to the endpoint B, keeping the angle with ABconstant, and remaining in the same plane as the pointsA, A′, and B.
Label the endpoint of the resulting segment B′ so that the segment is BB′.
Draw a straight line A′B′.
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, 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.
The length of this last geodesic constructed connecting the remaining points A′B′ 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 where terms of higher order in the length of the sides of the parallelogram have been suppressed.
Discrete approximation
can be discretely approximated by Schild's ladder, which approximates Levi-Civita parallelogramoids by approximate parallelograms.
