无目标
发表于 2025-3-23 12:15:34
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天文台
发表于 2025-3-23 14:10:53
Generalisations and applications,geometry in the plane are satisfied in sufficiently small regions. An inhabitant of such a world who always remains within some distance r of a fixed point (home, for example) could not detect in his world any contradictions to Euclidean plane geometry. But the real space in which we live is 3-dimen
octogenarian
发表于 2025-3-23 18:12:46
Geometries on the torus, complex numbers and Lobachevsky geometry,etry can be constructed as a geometry Σ. for a certain uniformly discontinuous group Γ of motions of the plane. It would seem that the classification of all such groups given in Chapter II, §8 then solves the problem. However, this is not quite the case: what we have done is to present a list of geo
acclimate
发表于 2025-3-23 23:02:27
Textbook 1994 the Euclidean plane or Euclidean 3-space. Starting from the simplest examples, we proceed to develop a general theory of such geometries, based on their relation with discrete groups of motions of the Euclidean plane or 3-space; we also consider the relation between discrete groups of motions and c
Limpid
发表于 2025-3-24 06:16:40
The theory of 2-dimensional locally Euclidean geometries,tries so obtained. On the other hand, this method turns out to be general enough to include any locally Euclidean geometry whatsoever, as will be proved in §10. This will then solve the problem of classifying all possible locally Euclidean geometries.
addition
发表于 2025-3-24 08:59:28
0172-5939 eometry of the Euclidean plane or Euclidean 3-space. Starting from the simplest examples, we proceed to develop a general theory of such geometries, based on their relation with discrete groups of motions of the Euclidean plane or 3-space; we also consider the relation between discrete groups of mot
keloid
发表于 2025-3-24 14:06:23
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走调
发表于 2025-3-24 15:04:05
Personal and Professional Alignments of Euclidean geometry in 3-space are satisfied in sufficiently small regions; we can think of the description of the 2-dimensional geometries as just a model for this more interesting problem. In this section, we will concern ourselves with the description and some of the properties of 3-dimensional locally Euclidean geometries.
antecedence
发表于 2025-3-24 22:30:39
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NEXUS
发表于 2025-3-25 00:56:52
Geometries on the torus, complex numbers and Lobachevsky geometry, belonging to the different Types I, II.a, II.b, III.a and III.b are different, since they are distinguished by properties such as the existence of closed curves, boundedness, and whether right and left are distinguishable. But it remains unclear whether the geometries within each type are distinct