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

【 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.