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We study the approach to gravity in which our curved spacetime is considered as a surface in a flat ambient space of higher dimension (the embedding theory). The dynamical variable in this theory is not a metric but an embedding function.…
We study homogeneous and isotropic cosmologies in a Weyl spacetime. We show that for homogeneous and isotropic spacetimes, the field equations can be reduced to the Einstein equations with a two-fluid source. We write the equations as a…
We establish a one-to-one correspondence between Finsler structures on the $2$-sphere with constant curvature $1$ and all geodesics closed on the one hand, and Weyl connections on certain spindle orbifolds whose symmetric Ricci curvature is…
A comprehensive approach to Sobolev-type embeddings, involving arbitrary rearrangement- invariant norms on the entire Euclidean space R^n, is offered. In particular, the optimal target space in any such embedding is exhibited. Crucial in…
Any Riemannian manifold has a canonical collection of valuations (finitely additive measures) attached to it, known as the intrinsic volumes or Lipschitz-Killing valuations. They date back to the remarkable discovery of H. Weyl that the…
Learning low-dimensional numerical representations from symbolic data, e.g., embedding the nodes of a graph into a geometric space, is an important concept in machine learning. While embedding into Euclidean space is common, recent…
We prove the existence of $C^{1,1}$ isometric immersions of several classes of metrics on surfaces $(\mathcal{M},g)$ into the three-dimensional Euclidean space $\mathbb{R}^3$, where the metrics $g$ have strictly negative curvature. These…
We apply the Weyl method, as sanctioned by Palais' symmetric criticality theorems, to obtain those -highly symmetric -geometries amenable to explicit solution, in generic gravitational models and dimension. The technique consists of…
We study the structure of isometries of the quadratic Wasserstein space $\mathcal{W}_2\left(\mathbb{S}^n,\|\cdot\|\right)$ over the sphere endowed with the distance inherited from the norm of $\mathbb{R}^{n+1}$. We prove that…
The group-theoretic method for constructing symmetric isometric embeddings is used to describe all possible four-dimensional surfaces in flat $(1,9)$-dimensional space, whose induced metric is static and spherically symmetric. For such…
This paper introduces a probabilistic formulation for the isometric embedding of a Riemannian manifold $(M^n,g)$ into Euclidean space $\mathbb{R}^q$. Given $\alpha \in ]\tfrac{1}{2},1]$, we show that a $C^{1,\alpha}$ embedding $u: M \to…
We develop the basic formulae of hyperspherical trigonometry in multidimensional Euclidean space, using multidimensional vector products, and their conversion to identities for elliptic functions. We show that the basic addition formulae…
We consider an embedding of a $2$-dimensional CW complex into the $3$-sphere, and construct it's dual graph. Then we obtain a homogeneous system of linear equations from the $2$-dimensional CW complex in the first homology group of the…
In this paper, we study the problem of local isometric immersion of pseudospherical surfaces determined by the solutions of a class of third order nonlinear partial differential equations with the type $u_t - u_{xxt} = \lambda u^2 u_{xxx} +…
We are concerned with underlying connections between fluids, elasticity, isometric embedding of Riemannian manifolds, and the existence of wrinkled solutions of the associated nonlinear partial differential equations. In this paper, we…
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…
In this paper, the Weierstrass technique for harmonic maps S^2 -> CP^(N-1) is employed in order to obtain surfaces immersed in multidimensional Euclidean spaces. It is shown that if the CP^(N-1) model equations are defined on the sphere S^2…
An explicit global and unique isometric embedding into hyperbolic 3-space, H^3, of an axi-symmetric 2-surface with Gaussian curvature bounded below is given. In particular, this allows the embedding into H^3 of surfaces of revolution having…
It is often easier to study pseudo-Riemannian manifolds by presenting them as surfaces in some ambient space. We propose an algorithm for construction of explicit isometric embeddings of pseudo-Riemannian manifolds with symmetries into an…
We discuss generalizations of the well-known theorem of Hilbert that there is no complete isometric immersion of the hyperbolic plane into Euclidean 3-space. We show that this problem is expressed very naturally as the question of the…