In a recent paper [1], the authors suggest a novel Riemannian framework for comparing shapes. In this framework, a simple closed surface is represented by a field of metric tensors and curvatures. A product Riemannian metric is developed based on the L norm on symmetric positive definite matrices and scalar fields. Taken as a quotient space under the group of volume-preserving diffeomorphisms, ...

We consider the problem of learning a Riemannian metric associated with a given differentiable manifold and a set of points. Our approach to the problem involves choosing a metric from a parametric family that is based on maximizing the inverse volume of a given dataset of points. From a statistical perspective, it is related to maximum likelihood under a model that assigns probabilities invers...

We prove that in metric measure spaces where the entropy functional is Kconvex along every Wasserstein geodesic any optimal transport between two absolutely continuous measures with finite second moments lives on a non-branching set of geodesics. As a corollary we obtain that in these spaces there exists only one optimal transport plan between any two absolutely continuous measures with finite ...

This paper will introduce a new way to compare parameterized surfaces using Riemannian shape analysis. To represent those surfaces, a form of q-maps will be developed which allow to define a special metric of functions that is invariant to rigid motion, global scaling and re-parametrization of the surfaces. This metric leads to the definition of the Riemannian distance function, which minimizes...

To motivate the construction, let us begin by looking at the much simpler situation that arises in real dimension 2. First of all, we all know that a complex curve (or Riemann surface) is the same thing as an oriented 2manifoldM equipped with a conformal class [g] of Riemannian (i.e. positivedefinite) metrics. If g is any Riemannian metric on M , and if u : M → R is any smooth positive function...

We study the differential geometric consequences of our previous result on the existence of fat triangulations, in conjunction with a result of Cheeger, Müller and Schrader, regarding the convergence of Lipschitz-Killing curvatures of piecewise-flat approximations of smooth Riemannian manifolds. A further application to the existence of quasiconformal mappings between manifolds, as well as an e...

We prove the statement in the title that is equivalent to the existence of a regular point of the sub-Riemannian exponential mapping. We also prove that the metric is analytic on an open everywhere dense subset in the case of a complete real-analytic sub-Riemannian mani-

This is the lecture notes on the interplay between optimal transport and Riemannian geometry. On a Riemannian manifold, the convexity of entropy along optimal transport in the space of probability measures characterizes lower bounds of the Ricci curvature. We then discuss geometric properties of general metric measure spaces satisfying this convexity condition. Mathematics Subject Classificatio...

In this note, we announce some results showing unexpected similarities between the moduli spaces of constant curvature metrics on 2-manifolds (the Riemann moduli space) and moduli spaces of Einstein metrics on 4manifolds. Let J? denote the moduli space of Einstein metrics of volume 1 on a compact, orientable 4-manifold M. If J£\ denotes the space of smooth Riemannian metrics of volume 1 on M, e...

BACKGROUND In Diffusion Tensor Imaging (DTI), Riemannian framework based on Information Geometry theory has been proposed for processing tensors on estimation, interpolation, smoothing, regularization, segmentation, statistical test and so on. Recently Riemannian framework has been generalized to Orientation Distribution Function (ODF) and it is applicable to any Probability Density Function (P...