具有伸缩结构的度量空间的无穷小affine geometry
Infinitesimal affine geometry of metric spaces endowed with a dilatation structure
摘要：
我们研究了具有伸缩结构的度量空间的代数和几何性质，这些伸缩结构是在通过越来越小的尺度时出现的。在极限中，我们得到了度量仿射几何的一个推广，赋予了一个非交换的向量加法运算和一个修正的共线三点比。这是赋范仿射群空间的几何学，一个包含齐次群、卡诺群或可收缩群的范畴。在这个范畴中，群运算不是基本的，而是派生的对象，并且仿射几何的推广不是基于关联关系。
英文摘要：
We study algebraic and geometric properties of metric spaces endowed with dilatation structures, which are emergent during the passage through smaller and smaller scales. In the limit we obtain a generalization of metric affine geometry, endowed with a noncommutative vector addition operation and with a modified version of ratio of three collinear points. This is the geometry of normed affine group spaces, a category which contains the ones of homogeneous groups, Carnot groups or contractible groups. In this category group operations are not fundamental, but derived objects, and the generalization of affine geometry is not based on incidence relations.
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