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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 side ''AB'') 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 (a geodesic). ==Construction== 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. * "Slide" (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''′. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Levi-Civita parallelogramoid」の詳細全文を読む スポンサード リンク
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