472:
17:
550:
Levi-Civita, Tullio (1917), "Nozione di parallelismo in una varietĂ qualunque e conseguente specificazione geometrica della curvatura riemanniana" [Notion of parallelism in any variety and consequent geometric specification of the
Riemannian curvature],
458:
333:
602:
553:
189:), 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
248:
669:
659:
197:, or that the construction is taking place in a suitable neighborhood, the steps to producing a Levi-Civita parallelogram are:
664:
598:"Nozione di parallelismo in una varietĂ qualunque e conseguente specificazione geometrica della curvatura riemanniana"
600:[Notion of parallelism in any variety and consequent geometric specification of the Riemannian curvature],
182:
300:
29:
453:{\displaystyle |A'B'|^{2}=|AB|^{2}+{\frac {8}{3}}\langle R(X,Y)X,Y\rangle +{\text{higher order terms}}}
572:
In the article by Levi-Civita (1917, p. 199), the segments AB and A'B ′ are called (respectively) the
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463:
where terms of higher order in the length of the sides of the parallelogram have been suppressed.
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41:
533:, which approximates Levi-Civita parallelogramoids by approximate parallelograms.
25:
16:
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The length of this last geodesic constructed connecting the remaining points
210:
80:′ will not in general be parallel to or the same length as the side
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85:
615:
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209:′. These geodesics are assumed to be parameterized by their
470:
15:
295:′ may in general be different than the length of the base
213:
in the case of a
Riemannian manifold, or to carry a choice of
177:, the notion of "straight line" generalizes to that of a
72:) and the same length as each other, but the fourth side
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constant, and remaining in the same plane as the points
336:
452:
173:or more generally any manifold equipped with an
64:′ of a parallelogramoid are parallel (via
307:′ be the exponential of a tangent vector
283:Quantifying the difference from a parallelogram
193:. Thus, assuming either that the manifold is
8:
603:Rendiconti del Circolo Matematico di Palermo
554:Rendiconti del Circolo Matematico di Palermo
439:
409:
303:. To state the relationship precisely, let
217:in the general case of an affine connection.
146:Label the endpoint of the resulting segment
505:are an approximation to first order of the
56:. Like a parallelogram, two opposite sides
251:. Label the endpoint of this geodesic by
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399:
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44:whose construction generalizes that of a
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299:. This difference is measured by the
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582:of the parallelogramoid in question.
319:the exponential of a tangent vector
104:Start with a straight line segment
52:. It is named for its discoverer,
529:can be discretely approximated by
108:and another straight line segment
14:
255:′, and the geodesic itself
243:The resulting tangent vector at
84:although it will be straight (a
645:, Math Sci Press, Massachusetts
150:′ so that the segment is
100:can be constructed as follows:
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20:Levi-Civita's parallelogramoid
1:
643:Geometry of Riemannian Spaces
247:generates a geodesic via the
169:In a curved space, such as a
596:Levi-Civita, Tullio (1917),
34:Levi-Civita parallelogramoid
686:
127:, keeping the angle with
301:Riemann curvature tensor
270:′ by the geodesic
187:normal coordinate system
670:Types of quadrilaterals
660:Curvature (mathematics)
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467:Discrete approximation
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201:Start with a geodesic
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665:Differential geometry
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205:and another geodesic
185:(such as a ball in a
157:Draw a straight line
30:differential geometry
19:
334:
262:Connect the points
171:Riemannian manifold
96:A parallelogram in
616:10.1007/BF03014898
527:Parallel transport
524:
507:parallel transport
450:
447:higher order terms
222:parallel transport
191:parallel transport
115:Slide the segment
98:Euclidean geometry
66:parallel transport
54:Tullio Levi-Civita
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181:. In a suitable
175:affine connection
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215:affine parameter
123:to the endpoint
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531:Schild's ladder
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50:Euclidean plane
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606:(in Italian),
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226:tangent vector
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119:′ along
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475:Two rungs of
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46:parallelogram
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38:quadrilateral
35:
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27:
18:
642:
639:Cartan, Élie
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183:neighborhood
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109:
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95:
92:Construction
81:
77:
73:
69:
61:
60:′ and
57:
42:curved space
33:
26:mathematical
23:
610:: 173–205,
68:along side
654:Categories
624:46.1125.02
590:References
632:122088291
580:suprabase
440:⟩
410:⟨
220:"Slide" (
211:arclength
28:field of
641:(1983),
355:′
347:′
278:′.
259:′.
195:complete
179:geodesic
165:′.
154:′.
112:′.
86:geodesic
327:. Then
291:′
274:′
161:′
76:′
48:in the
24:In the
630:
622:
315:, and
224:) the
32:, the
628:S2CID
561:: 199
537:Notes
40:in a
36:is a
578:and
575:base
492:and
620:JFM
612:doi
509:of
323:at
311:at
236:to
228:of
88:).
82:AB,
656::
626:,
618:,
608:42
559:42
317:AB
305:AA
297:AB
257:BB
230:AA
207:AA
203:AB
152:BB
135:,
129:AB
121:AB
117:AA
110:AA
106:AB
70:AB
62:BB
58:AA
614::
563:.
520:0
517:X
514:0
511:A
503:2
500:X
497:2
494:A
490:1
487:X
484:1
481:A
443:+
437:Y
434:,
431:X
428:)
425:Y
422:,
419:X
416:(
413:R
405:3
402:8
397:+
392:2
387:|
382:B
379:A
375:|
371:=
366:2
361:|
352:B
344:A
339:|
325:A
321:Y
313:A
309:X
293:B
289:A
276:B
272:A
268:B
264:A
253:B
245:B
240:.
238:B
234:A
163:B
159:A
148:B
143:.
141:B
137:A
133:A
125:B
78:B
74:A
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