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Streamlines of a relativistic perfect isentropic fluid are geodesics of a Riemannian space whose metric is defined by enthalpy of the fluid. This fact simplifies the solution of some problems, as is also of interest from the point of view…

We develop a geometric formulation of fluid dynamics, valid on arbitrary Riemannian manifolds, that regards the momentum-flux and stress tensors as 1-form valued 2-forms, and their divergence as a covariant exterior derivative. We review…

Geodesic problems involve computing trajectories between prescribed initial and final states to minimize a user-defined measure of distance, cost, or energy. They arise throughout physics and engineering -- for instance, in determining…

[1] investigates advanced connotations of Hardy and Rellich-type inequalities on complete noncompact Riemannian manifolds, delving on deriving inequalities that incorporate poignant weight functions. These inequalities prolongate classical…

This is a short presentation of some classical results on finite dimensional complex Lie algebras (classification of nilpotent Lie algebras, deformations and perturbations, contractions and rigidity). We present some applications to…

Algorithms for the computation of the forward and inverse geodesic problems for an ellipsoid of revolution are derived. These are accurate to better than 15 nm when applied to the terrestrial ellipsoids. The solutions of other problems…

Any three-dimensional Riemannian metric can be locally obtained by deforming a constant curvature metric along one direction. The general interest of this result, both in geometry and physics, and related open problems are stressed.

In this work, we propose to study the global geometrical properties of generative models. We introduce a new Riemannian metric to assess the similarity between any two data points. Importantly, our metric is agnostic to the parametrization…

This paper reviews several Riemannian metrics and evolution equations in the context of diffeomorphic shape analysis. After a short review of of various approaches at building Riemannian spaces of shapes, with a special focus on the…

A survey of some recent and important results which have to do with integrable equations and their relationship with the theory of surfaces is given. Some new results are also presented. The concept of the moving frame is examined, and it…

We present alternative postulates for Euclidean geometry whose merit is that they lead to a new class of invariants and associated geometries for real finite-dimensional unital associative algebras.

In this article we will represent some ideas and a lot of new theorems in Euclidean plane geometry.

We survey recent results on inverse problems for geodesic X-ray transforms and other linear and non-linear geometric inverse problems for Riemannian metrics, connections and Higgs fields defined on manifolds with boundary.

Manifolds discovered by machine learning models provide a compact representation of the underlying data. Geodesics on these manifolds define locally length-minimising curves and provide a notion of distance, which are key for reduced-order…

We use the method of synthetic differential geometry to revisit the geometric reasoning employed by Lie, Klein and others in their study of partial differential equations.

We investigate the geometrical structure of probabilistic generative dimensionality reduction models using the tools of Riemannian geometry. We explicitly define a distribution over the natural metric given by the models. We provide the…

We study the interplay between geometry and partial differential equations. We show how the fundamental ideas we use require the ability to correctly calculate the dimensions of spaces associated to the varieties of zeros of the symbols of…

We suggest a construction that, given a trajectorial diffeomorphism between two Hamiltonian systems, produces integrals of them. As the main example we treat geodesic equivalence of metrics. We show that the existence of a non-trivially…

Non-Euclidean method of the generalized geometry construction is considered. According to this approach any generalized geometry is obtained as a result of deformation of the proper Euclidean geometry. The method may be applied for…

Understanding, finding, or even deciding on the existence of real solutions to a system of equations is a very difficult problem with many applications. While it is hopeless to expect much in general, we know a surprising amount about these…