Related papers: On the isometric version of Whitney's strong embed…

It is shown that in dimension at least three a local diffeomorphism of Euclidean n-space into itself is injective provided that the pull-back of every plane is a Riemannian submanifold which is conformal to a plane. Using a similar…

Let $S$ be an oriented closed surface of genus at least two, and let $M = S \times (0,1)$. Suppose that $h$ is a Riemannian metric on $S$ with curvature strictly greater than $-1$, $h^{*}$ is a Riemannian metric on $S$ with curvature…

In this paper we give a proof of Lichnerowicz Conjecture for compact simply connected manifolds which is intrinsic in the sense that it avoids the {\it Nice Embeddings} into eigen spaces of the Laplacian. Even if one wants to use these…

We prove that the universal cover of any graph manifold quasi-isometrically embeds into a product of three trees. In particular we show that the Assouad-Nagata dimension of the universal cover of any closed graph manifold is 3, proving a…

We generalize Cauchy's celebrated theorem on the global rigidity of convex polyhedra in Euclidean $3$-space $\mathbb{E}^{3}$ to the context of circle polyhedra in the $2$-sphere $\mathbb{S}^{2}$. We prove that any two convex and proper…

We study how to construct explicit deformations of generic smooth maps from closed $n$--dimensional manifolds $M$ with $n \geq 2$ to the $2$--sphere $S^2$ and show that every smooth map $M \to S^2$ is homotopic to a $C^\infty$ stable map…

The isometric immersion of two-dimensional Riemannian manifold with negative Gauss curvature into the three-dimensional Euclidean space is considered through the Gauss-Codazzi equations for the first and second fundamental forms. The large…

We develop a new approach to the classical problem on isotopy classification of embeddings of manifolds into Euclidean spaces. This approach involves studying of a new embedding invariant, of almost-embeddings and of smoothing, as well as…

We explore the practicability of Nash's Embedding Theorem in vision and imaging sciences. In particular, we investigate the relevance of a result of Burago and Zalgaller regarding the existence of isometric embeddings of polyhedral surfaces…

We prove that if an asymptotically Schwarzschildean 3-manifold (M,g) contains a properly embedded stable minimal surface, then it is isometric to the Euclidean space. This implies, for instance, that in presence of a positive ADM mass any…

The purpose of this article is to study co-dimension $2$ iso-contact embeddings of closed contact manifolds. We first show that a closed contact manifold $(M^{2n-1}, \xi_M)$ iso-contact embeds in a contact manifold $(N^{2n+1}, \xi_N),$…

Let $K$ be a nontrivial knot. For each $n\in \mathbb{N}$, we prove that the rank of its $n$th iterated Whitehead doubled knot group $\pi_1(S^3 \setminus \operatorname{WD}^n(K))$ is bounded below by $n+1$. As an application, we show that…

In the special case of S^1 invariant metrics on S^2, we find necessary and sufficient conditions for the existence of isometric embeddings into the canonical R^3, in other words: a Weyl type theorem with converse.

We prove that if $X$ is a topological space that admits Debreu's classical utility theorem (eg.\ $X$ is separable and connected, second countable, etc.), then order relations on $X$ satisfying milder completeness conditions can be…

We prove a universal lower bound for the $L^{n/2}$-norm of the Weyl tensor in terms of the Betti numbers for compact $n$-dimensional Riemannian manifolds that are conformally immersed as hypersurfaces in the Euclidean space. As a…

We say that a topological $n$-manifold $N$ is a cubical $n$-manifold if it is contained in the $n$-skeleton of the canonical cubulation $\mathcal{C}$ of ${\mathbb{R}}^{n+k}$ ($k\geq1$). In this paper, we prove that any closed, oriented…

For a compact connected Lie group $G$ acting as isometries on a compact orientable Riemannian manifold $M^{n+1},$ and cohomogeneity not equal to 0 or 2, we prove the existence of a nontrivial embedded $G$-invariant minimal hypersurface,…

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…

Totally geodesically embeddings of infinitely many closed 7-manifolds into 13-dimensional positively curved closed Riemannian manifolds are constructed. The problems of computing pinching constants and existence of other totally geodesical…

In this paper, we study the coarse embedding into Banach space. We proved that under certain conditions, the property of embedding into Banach space can be preserved under taking the union the metric spaces. For a group $G$ strongly…