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Abstract: We prove that a Ricci curvature based method of triangulation of compactRiemannian manifolds, due to Grove and Petersen, extends to the context ofweighted Riemannian manifolds and more general metric measure spaces. In bothcases the role of the lower bound on Ricci curvature is replaced by thecurvature-dimension condition ${ m CD}K,N$. We show also that for weightedRiemannian manifolds the triangulation can be improved to become a thick oneand that, in consequence, such manifolds admit weight-sensitivequasimeromorphic mappings. An application of this last result to informationmanifolds is considered.Further more, we extend to weak ${ m CD}K,N$ spaces the results of Kanairegarding the discretization of manifolds, and show that the volume growth ofsuch a space is the same as that of any of its discretizations.



Autor: Emil Saucan

Fuente: https://arxiv.org/



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