Related papers: Two-Scale Geometric Modelling for Defective Media

It is known that the only finite-dimensional diffeological vector space that admits a diffeologically smooth scalar product is the standard space of appropriate dimension. In this note we consider a way to circumnavigate this issue, by…

To any metric spaces there is an associated metric profile. The rectifiability of the metric profile gives a good notion of curvature of a sub-Riemannian space. We shall say that a curvature class is the rectifiability class of the metric…

In Riemannian geometry geodesics are integral curves of the Riemannian distance gradient. We extend this classical result to the framework of Information Geometry. In particular, we prove that the rays of level-sets defined by a…

We define a regularized size-shape distortion (quality) measure for curved high-order elements on a Riemannian space. To this end, we measure the deviation of a given element, straight-sided or curved, from the stretching, alignment, and…

This work introduces the Grassmannian Diffusion Maps, a novel nonlinear dimensionality reduction technique that defines the affinity between points through their representation as low-dimensional subspaces corresponding to points on the…

Models of discrete space and space-time that exhibit continuum-like behavior at large lengths could have profound implications for physics. They may tame the infinities that arise from quantizing gravity, and dispense with the machinery of…

A statistical manifold is a pseudo-Riemannian manifold endowed with a Codazzi structure. This structure plays an important role in Information Geometry and its related fields, e.g., a statistical model admits this structure with the…

A Riemannian metric bundle G(M) is a fiber bundle over a smooth manifold M, whose fibers are the spaces of symmetric, positive-definite bilinear forms on the tangent spaces of M, which represent the Rieman?nian metrics. In this work, we aim…

One of the most appealing results of metric-affine gauge theory of gravity is a close parallel between the Riemann curvature two-form and the Cartan torsion two-form: While the former is the field strength of the Lorentz-group connection…

We begin by studying the Riemannian geometry of the tangent Lie group $TG$ associated with a Lie group $G$ whose commutator subgroup is two-dimensional, equipped with the lift of a left-invariant Riemannian metric on $G$. We establish the…

Upon a consistent topological statistical theory the application of structural statistics requires a quantification of the proximity structure of model spaces. An important tool to study these structures are Pseudo-Riemannian metrices,…

Deep generative models are universal tools for learning data distributions on high dimensional data spaces via a mapping to lower dimensional latent spaces. We provide a study of latent space geometries and extend and build upon previous…

This paper proposes an original Riemmanian geometry for low-rank structured elliptical models, i.e., when samples are elliptically distributed with a covariance matrix that has a low-rank plus identity structure. The considered geometry is…

Let $g_t$ be a smooth 1-parameter family of negatively curved metrics on a closed hyperbolic 3-manifold $M$ starting at the hyperbolic metric. We construct foliations of the Grassmann bundle $Gr_2(M)$ of tangent 2-planes whose leaves are…

Within the framework of Relativistic Schroedinger Theory (an alternative form of quantum mechanics for relativistic many-particle systems) it is shown that a general N-particle system must occur in one of two forms: either as a ``positive''…

This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive…

We consider invariant Riemannian metrics on compact homogeneous spaces G/H where an intermediate subgroup K between G and H exists, so that the homogeneous space G/H is the total space of a Riemannian submersion. We study the question as to…

In a geometrical approach to gravity the metric and the (gravitational) connection can be independent and one deals with metric-affine theories. We construct the most general action of metric-affine effective field theories, including a…

The configuration space of general relativity is superspace - the space of all Riemannian 3-metrics modulo diffeomorphisms. However, it has been argued that the configuration space for gravity should be conformal superspace - the space of…

Diffusion models are powerful deep generative models, but unlike classical models, they lack an explicit low-dimensional latent space that parameterizes the data manifold. This absence makes it difficult to perform manifold-aware…