Related papers: Isometric Embeddings of Polyhedra into Euclidean S…

We prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash $C^1$ Embedding Theorem. For more general metric spaces the same…

We present an explicit piecewise linear map from a flat Klein bottle (i.e. one that is locally isometric to the Euclidean plane) into Euclidean 3-space an that is an isometric immersion -- a path isometry that is locally injective. The…

This paper is devoted to investigating the isometric immersion problem of Riemannian manifolds in a high codimension. It has recently been demonstrated that any short immersion from an $n$-dimensional smooth compact manifold into…

We prove that if an RCD space has a regular isometric immersion in a Euclidean space, then the immersion is a locally bi-Lipschitz embedding map. This result leads us to prove that if a compact non-collapsed RCD space has an isometric…

We present numerical polyhedron data for the image of a piecewise-linear map from a zero-curvature Klein bottle into Euclidean 3-space such that every point in the domain has a neighborhood which is isometrically embedded. To the author's…

A proof of the isometric embedding of a given two-metric in E^3 of class C^1. The method uses the theory of first order partial differential equations. The curvature of the metric plays no role in the proof.

This paper is concerned with embeddings of homogeneous spaces into Euclidean spaces. We show that any homogeneous metric space can be embedded into a Hilbert space using an almost bi-Lipschitz mapping (bi-Lipschitz to within logarithmic…

We extend the results of B. Minemyer by showing that any indefinite metric polyhedron (either compact or not) with the vertex degree bounded from above admits an isometric simplicial embedding into a Minkowski space of the lowest possible…

A result due to Burago and Zalgaller (1960, 1995) states that every orientable polyhedral surface, one that is obtained by gluing Euclidean polygons, has an isometric piecewise linear (PL) embedding into Euclidean space $\mathbb{E}^3$. A…

A metric polygon is a metric space comprised of a finite number of closed intervals joined cyclically. The second-named author and Ntalampekos recently found a method to bi-Lipschitz embed an arbitrary metric triangle in the Euclidean plane…

We give a sufficient condition for a projective metric on a subset of a Euclidean space to admit a bi-Lipschitz embedding into Euclidean space of the same dimension.

Although the Nash theorem solves the isometric embedding problem, matters are inherently more involved if one is further seeking an embedding that is well-behaved from the standpoint of submanifold geometry. More generally, consider a…

The Nash-Kuiper Theorem states that the collection of $C^1$-isometric embeddings from a Riemannian manifold $M^n$ into $\mathbb{E}^N$ is $C^0$-dense within the collection of all smooth 1-Lipschitz embeddings provided that $n < N$. This…

Petrunin proves that a metric space $\mathcal{X}$ admits an intrinsic isometry into $\mathbb{E}^n$ if and only if $\mathcal{X}$ is a pro-Euclidean space of rank at most $n$. He then shows that either case implies that $\mathcal{X}$ has…

In [The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces. Memoirs of the American Mathematical Society. American Mathematical Society, 2023], Sturm studied the space of all metric measure spaces up…

We prove that the isometric embedding of any metric of differentiability class C1 in E3 exists. We use simplified notation for the given metric, namely geodesic parameters, and level parameters for the embedded surface in E3. Central to our…

We prove that every proper $n$-dimensional length metric space admits an "approximate isometric embedding" into Lorentzian space $\mathbb{R}^{3n+6,1}$. By an "approximate isometric embedding" we mean an embedding which preserves the energy…

We prove a version of Whitney's strong embedding theorem for isometric embeddings within the general setting of the Nash-Kuiper h-principle. More precisely, we show that any $n$-dimensional smooth compact manifold admits infinitely many…

We construct an isometric embedding of a bounded set in a Euclidean space into the Gromov-Hausdorff space. In particular, we can embed a bounded and connected Riemannian manifold into the Gromov-Hausdorff space by a bilipschitz map.

Consider the sum of the first $N$ eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for $N$ sufficiently large the map is an embedding. In analogy with a fruitful idea of…