Quantifying The Difference From A Parallelogram
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.
Read more about this topic: Levi-Civita Parallelogramoid
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“Man is the only animal that laughs and weeps; for he is the only animal that is struck with the difference between what things are and what they ought to be.”
—William Hazlitt (1778–1830)