Riemannian geometry of compact metric spaces
Palmer, Ian Christian
MetadataShow full item record
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.