Curvature based triangulation of metric measure spaces

Abstract

We prove that a Ricci curvature based method of triangulation of compact Riemannian manifolds, due to Grove and Petersen, extends to the context of weighted Riemannian manifolds and more general metric measure spaces. In both cases the role of the lower bound on Ricci curvature is replaced by the curvature-dimension condition CD(K, N ). We show also that for weighted Riemannian manifolds the triangulation can be improved to become a thick one and that, in consequence, such manifolds admit weight-sensitive quasimeromorphic mappings. An application of this last result to information manifolds is considered.Further more, we extend to weak CD(K, N ) spaces the results of Kanai regarding the discretization of manifolds, and show that the volume growth of such a space is the same as that of any of its discretizations.1 2 EMIL SAUCAN without any modifications to metric measure spaces (except, of course, the obvious necessary adaptations to the more general context).The reason behind this is dual: On one hand there exists a feeling in the community, that, although the tools and results developed by Gromov, Lott, Villani, Sturm and others are most elegant, they lack, so far, any concrete and efficient application. We wish, therefore, to further emphasize that the notions of curvatures for metric measure spaces are highly natural by giving an extension of a classical problem and its (also classical) solution, to the new context. On the other hand, it appears that there exists a real interest in the triangulation and representation of the information manifold, that is the space of parameterized probability measures (or the statistical model) equipped with the Riemannian metric induced by the Fischer information metric onto the Euclidean sphere (see Section 4.1.3 below).The reminder of this paper is structured as follows: As already mentioned above, we present, in Section 2, the construction of Grove and Petersen. (This and the following section are the most "didactic", thus we proceed rather slowly; however, afterwards the pace of the exposition will become more brisk.) Next, in Section 3, we introduce the necessary notions and results regarding curvatures of metric measure spaces. (Unfortunately, the style is rather technical, since we had to cover quite a large number of definitions and results.) Section 4 constitutes the heart of our paper, in the sense that we show therein how to extend Grove-Petersen construction to the metric measure setting. We follow, in Sections 5, with an extension, to weak CD(K, N ) spaces, of Kanai's results regarding the discretization of manifolds, and show, in particular, that the volume growth of such a space coincides with that of any of its discretizations. In a sense, this represents the second part of the present paper, related to, yet distinct, from the main triangulation problem considered in the previous sections. We conclude, in Section 6, with a few very brief remarks.

Fat Triangulations, Curvature and Quasiconformal Mappings

“…5 It follows that Theorems 1.1 generalizes to almost Riemannian manifolds, and we formalize this observation as Remark 1.19. Besides the slight generalization (and its applications) presented above, there exist other, perhaps more less immediate ones -see [Sa08a], [Sa11a]. In particular, we noted the existence of fat triangulations of Lipschitz manifolds.…”

“…Indeed, while volume and edge lengths of a Euclidean simplex (or, for that matter, in any space form) are closely related, in a general metric space no volume is apriorily postulated. Admittedly, one can attempt a fitting definition for metric measure spaces [Sa11a], but this has only limited purely mathematical applications and, furthermore, it also departs from the purely metric approach. 7 Fortunately, this problem is easy to amend, using the so called Cayley-Menger determinant, that expresses the volume of the n-dimensional Euclidean simplex σ n (p 0 , p 1 , .…”

Generalized Ricci curvature based sampling and reconstruction of images

“…• For compact manifolds (such as appearing in Graphics) use the more precise sampling density provided in [14] and extended to weighted manifolds in [15].…”

“…In the papers mentioned above [14][15][16], the sampling density is theoretically, essentially, equal to 1/| max Ric ϕ |. (This is a generalization of the similar sampling criterion using Gaussian/sectional curvature for surfaces [5] and higher dimensional manifolds [14].)…”

“…In fact, Grove and Petersen also proved in [14] a stronger result, namely that Ricci curvature (which is a weaker invariant than Gaussian curvature) also suffices to topologically reconstruct the manifold (up to homotopy type, to be more precise). This result was extended to manifolds with density (weighted manifolds) in [15], where the role of Ricci curvature is played by a generalized Ricci curvature, that includes, as a special case, the one given by (1). This approach was shown in [16] to be a proper sampling method, allowing for good reconstruction of the manifold from the metric, not just topological, viewpoint.…”