学位论文详细信息
Riemannian geometry of compact metric spaces
Noncommutative Geometry;Metric Spaces
Palmer, Ian Christian ; Mathematics
University:Georgia Institute of Technology
Department:Mathematics
关键词: Noncommutative Geometry;    Metric Spaces;   
Others  :  https://smartech.gatech.edu/bitstream/1853/34744/1/palmer_ian_c_201008_phd.pdf
美国|英语
来源: SMARTech Repository
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【 摘 要 】

A construction is given for which the Hausdorff measure and dimension of an arbitrary abstract compact metric space (X, d) can be encoded in a spectral triple. By introducing the concept of resolving sequence of open covers, conditions are given under which the topology, metric, and Hausdorff measure can be recovered from a spectral triple dependent on such a sequence. The construction holds for arbitrary compact metric spaces, generalizing previous results for fractals, as well as the original setting of manifolds, and also holds when Hausdorff and box dimensions differ---in particular, it does not depend on any self-similarity or regularity conditions on the space. The only restriction on the space is that it have positive s₀ dimensional Hausdorff measure, where s₀ is the Hausdorff dimension of the space, assumed to be finite. Also, X does not need to be embedded in another space, such as Rⁿ.

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