Euclidean geometry的不同表示及其在时空几何中的应用
Different representations of Euclidean geometry and their application to the space-time geometry
摘要：
三种不同的表示适当的欧几里德几何被认为是。它们在基本元素的数量上有所不同，从这些基本元素中构造出几何对象。在E-表示中，有三个基本元素（点、线段、角），没有额外的结构。V-表示包含两个基本元素（点，向量）和附加结构：线性向量空间。在sigma表示中，只有一个基本元素和附加结构：世界函数σ=ρ^{2}/2，其中ρ是距离。距离的概念出现在所有表示中。然而，作为一个结构，确定几何形状，距离只出现在西格玛表示。sigma表示最适合于修改适当的欧几里德几何。实际上，适当的欧几里德几何的任何修改都将其变成多变量几何，其中存在许多向量Q_0Q_1，Q_0Q_1^{ \prime }，.，它们等于矢量P_0P_1，但它们之间一般不相等。多元概念在时空几何中的应用是非常重要的。真实的时空几何是多元的。时空几何的多元性是量子效应的原因。
英文摘要：
Three different representation of the proper Euclidean geometry are considered. They differ in the number of basic elements, from which the geometrical objects are constructed. In E-representation there are three basic elements (point, segment, angle) and no additional structures. V-representation contains two basic elements (point, vector) and additional structure: linear vector space. In sigma-representation there is only one basic element and additional structure: world function σ=ρ^{2}/2, where ρis the distance. The concept of distance appears in all representations. However, as a structure, determining the geometry, the distance appears only in the sigma-representation. The sigma-representation is most appropriate for modification of the proper Euclidean geometry. Practically any modification of the proper Euclidean geometry turns it into multivariant geometry, where there are many vectors Q_0Q_1, Q_0Q_1^{\prime},..., which are equal to the vector P_0P_1, but they are not equal between themselves, in general. Concept of multivariance is very important in application to the space-time geometry. The real space-time geometry is multivariant. Multivariance of the space-time geometry is responsible for quantum effects.
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