Levi-Civita parallelogramoid

In the mathematical field of differential geometry, the Levi-Civita parallelogramoid is a quadrilateral[1] in a curved space whose construction generalizes that of a parallelogram in the Euclidean plane.

[2] 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: 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.