Levi-Civita parallelogramoid

In the mathematical field of differential geometry, the Levi-Civita parallelogramoid is a certain figure generalizing a parallelogram to a curved space. It is named for its discoverer, Tullio Levi-Civita. 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 "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 (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.
* 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 "A"′"B"′.

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

:$|A\text{'}B\text{'}|^2\; =\; |AB|^2\; +\; frac\{8\}\{3\}langle\; R(X,Y)X,Y\; angle\; +\; h.o.t.$

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

ee also

* Schild's ladder

References

*citation | first = Elie | last = Cartan | authorlink=Elie Cartan|title = Geometry of Riemannian Spaces| publisher = Math Sci Press, Massachusetts | year = 1983