Related papers: Nontrivial isometric embeddings for flat spaces

Metric embedding has become a common technique in the design of algorithms. Its applicability is often dependent on how high the embedding's distortion is. For example, embedding finite metric space into trees may require linear distortion…

For a fixed $K\gg 1$ and $n\in\mathbb{N}$, $n\gg 1$, we study metric spaces which admit embeddings with distortion $\le K$ into each $n$-dimensional Banach space. Classical examples include spaces embeddable into $\log n$-dimensional…

To what extent are two images picturing the same 3D surfaces? Even when this is a known scene, the answer typically requires an expensive search across scale space, with matching and geometric verification of large sets of local features.…

In this paper, isometric embedding results of Greene, Gromov and Rokhlin are extended to what are called "indefinite metric polyhedra". Two definitions for an indefinite metric polyhedron are given, an intuitive definition and a more useful…

We consider the isometric deformation problem for oriented non simply connected immersed minimal surfaces $f:M \to S^{4}$. We prove that the space of all isometric minimal immersions of $M$ into $S^{4}$ with the same normal curvature…

The isometric embedding of surfaces in three-dimensional space is fundamental to various physical systems, from elastic sheets to programmable materials. While continuous surfaces typically admit unique solutions under suitable boundary…

We investigate the use of Minimax distances to extract in a nonparametric way the features that capture the unknown underlying patterns and structures in the data. We develop a general-purpose and computationally efficient framework to…

A smooth map between smooth manifolds is called a special generic map if it has only definite fold points as its singularities. In this paper, we give conditions for a special generic map into the 3-dimensional Euclidean space to be…

Let $\varphi:F_1\to F_2$ be an injective morphism of free groups. If $\varphi$ is geometric (i.e. induced by an inclusion of oriented compact connected surfaces with nonempty boundary), then we show that $\varphi$ is an isometric embedding…

Motivated by well-known obstacles to quantum gravity, I look for the most general geometrodynamical symmetries compatible with a reduced physical configuration space for metric gravity. I argue that they lead either to a completely static…

We initiate the study of metric embeddings with \emph{outliers}. Given some metric space $(X,\rho)$ we wish to find a small set of outlier points $K \subset X$ and either an isometric or a low-distortion embedding of $(X\setminus K,\rho)$…

We obtain a dynamical--topological obstruction for the existence of isometric embedding of a Riemannian manifold-with-boundary $(M,g)$: if the first real homology of $M$ is nontrivial, if the centre of the fundamental group is trivial, and…

This note pertains to isometric embeddings endowed with certain geometric properties. We study two embedding problems for a Riemannian manifold $M$ which is diffeomorphic to $\RR^n$ and admits a Bieberbach group $\Gamma$ acting by…

General relativity differs from other forces in nature in that it can be made to disappear locally. This is the essence of the equivalence principle. In general relativity the equivalence principle is implemented using differential…

A consistent set of asymptotic conditions for the simplest supergravity theory without cosmological constant in three dimensions is proposed. The canonical generators associated to the asymptotic symmetries are shown to span a…

The finiteness requirement for Euclidean Einstein gravity is shown to be so stringent that only the flat metric is allowed. We examine counterterms in 4D and 6D Ricci-flat manifolds from general invariance arguments.

We construct the differential geometry of smooth manifolds equipped with an algebraic curvature map acting as an area measure. Area metric geometry provides a spacetime structure suitable for the discussion of gauge theories and strings,…

Certain semi-Riemannian metrics can be decomposed into a Riemannian part and an isochronal part. The properties of such metrics are particularly easy to visualize in a coordinate-free way, using isometric embedding. We present such an…

The analysis of manifold valued data using embedding based methods is linked to the problem of finding suitable embeddings. In this paper we are interested in embeddings of quotient manifolds $\mathrm{SO}(3)/\mathcal{S}$ of the rotation…

We make biframe and quaternion extensions on the noncommutative geometry, and construct the biframe spacetime for the unification of gravity and quantum field theory. The extended geometry distinguishes between the ordinary spacetime based…