Related papers: Isometric immersions and applications
This paper studies isometric immersions of space forms by means of a hierarchy of finite dimensional integrable systems in Lax form on loop algebras.
The isometric embedding problem is a fundamental problem in differential geometry. A longstanding problem is considered in this paper to characterize intrinsic metrics on a two-dimensional Riemannian manifold which can be realized as…
A very brief introduction to tropical and idempotent mathematics is presented. Applications to classical mechanics and geometry are especially examined.
Using the results of \cite{P1}, we get some estimates of warping functions for isometric immersions by changing the target manifolds by some types of Riemannian manifolds: constant space forms and Hermitian symmetric spaces. And we deal…
We provide conditions under which an isometric immersion of a (warped) product of manifolds into a space form must be a (warped) product of isometric immersions.
In this article we propose a novel geometric model to study the motion of a physical flag. In our approach a flag is viewed as an isometric immersion from the square with values in $\mathbb R^3$ satisfying certain boundary conditions at the…
Geometric mechanics is usually studied in applied mathematics and most introductory texts are hence aimed at a mathematically minded audience. The present note tries to provide the intuition of geometric mechanics and to show the relevance…
Some examples and basic properties of ultrametric spaces are briefly discussed.
A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold ${\mathcal M}^2$ which can be realized as isometric immersions into $\R^3$. This problem can be formulated as…
Manifold learning has been proven to be an effective method for capturing the implicitly intrinsic structure of non-Euclidean data, in which one of the primary challenges is how to maintain the distortion-free (isometry) of the data…
An isometric embedding of a graph into a metric space is an embedding of the vertices such that the smallest number of edges connecting any two vertices equals to the distance in the metric space between the images. In this paper, we study…
This paper is about Information Geometry, a relatively new subject within mathematical statistics that attempts to study the problem of inference by using tools from modern differential geometry. This paper provides an overview of some of…
A very elementary introduction to quantum algebras is presented and a few examples of their physical applications are mentioned.
This paper deals with the famous isoperimetric inequality. In a first part, we give some new functional form of the isoperimetric inequality, and in a second part, we give a quantitative form with a remainder term involving Wasserstein…
In this paper, we introduce the notion of developments of curves with respect to symmetric tensors and use it to prove the existence of isometric immersions into a general ambient space with prescribed second fundamental form. Our method…
The purpose of this paper is to give, on one hand, a mathematical exposition of the main topological and geometrical properties of geometric transitions, on the other hand, a quick outline of their principal applications, both in…
The isometric immersion of two-dimensional Riemannian manifolds or surfaces in the three-dimensional Euclidean space is a fundamental problem in differential geometry. When the Gauss curvature is negative, the isometric immersion problem is…
We study the Darboux equation, a fundamental PDE arising in the theory of isometric immersions of two-dimensional Riemannian manifolds into $\mathbb{R}^3$, in the low-regularity regime. We introduce a notion of weak solution for $u\in…
This is an expository article for Elsevier's Encyclopedia of Mathematical Physics on the subject in the title. Comments/corrections welcome.
This paper presents a brief but comprehensive introduction to certain mathematical techniques in General Relativity. Familiar mathematical procedures are investigated taking into account the complications of introducing a non trivial…