Related papers: Canonical isometric embeddings of projective space…

For any n-dimensional compact Riemannian manifold (M,g), we construct a canonical t-family of isometric embeddings I_{t}: M->R^{q(t)}, with t>0 sufficiently small and q(t)>>t^{-n/2}. This is done by intrinsically perturbing the heat kernel…

We study the problem of construction of explicit isometric embeddings of (pseudo)-Riemannian manifolds. We discuss the method which is based in the idea that the exterior symmetry of the embedded surface and the interior symmetry of the…

For an ideal $I_{m,n}$ generated by all square-free monomials of degree $m$ in a polynomial ring $R$ with $n$ variables, we obtain a specific embedding of a canonical module of $R/I_{m,n}$ to $R/I_{m,n}$ itself. The construction of this…

A recursive construction is presented for the projective cubature formulas of index $p$ on the unit spheres $S(m,K)\subset K^m$ where $K$ is $R$ or $C$, or $H$. This yields a lot of new upper bounds for the minimal number of nodes…

Let $S$ denote the unit sphere of a real normed space. We show that the intrinsic metric on $S$ is strongly equivalent to the induced metric on $S$. Specifically, for all $x,y\in S$, \[ \|x-y\|\leq d(x,y)\leq\sqrt{2}\pi\|x-y\|, \] where $d$…

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 we study critial isometric and minimal isometric embeddings of classes of Riemannian metrics which we call {\it quasi-$k$-curved metrics}. Quasi-$k$-curved metrics generalize the metrics of space forms. We construct explicit…

We present a numerical method for solving Weyl's embedding problem which consists of finding a global isometric embedding of a positively curved and positive-definite spherical 2-metric into the Euclidean three space. The method is based on…

Photometric Stereo methods seek to reconstruct the 3d shape of an object from motionless images obtained with varying illumination. Most existing methods solve a restricted problem where the physical reflectance model, such as Lambertian…

Let $E$ be a smooth bundle with fiber an $n$-dimensional real projective space $\mathbb{R}P^n$. We show that, if every fiber carries a positively curved pointwise strongly $1/4$-pinched Riemannian metric that varies continuously with…

We give a new proof for the local existence of a smooth isometric embedding of a smooth $3$-dimensional Riemannian manifold with nonzero Riemannian curvature tensor into $6$-dimensional Euclidean space. Our proof avoids the sophisticated…

For a given Hilbert space $\mathcal H$, consider the space of self-adjoint projections $\mathcal P(\mathcal H)$. In this paper we study the differentiable structure of a canonical sphere bundle over $\mathcal P(\mathcal H)$ given by $$…

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…

We give a sufficient condition for branched minimal immersions of spheres into ellipsoids to be embedded: we show that if the coordinate functions of the branched minimal immersion are first or second eigenfunctions with respect to a…

We investigate a relations of almost isometric embedding and almost isometry between metric spaces and prove that with respect to these relations: (1) There is a countable universal metric space. (2) There may exist fewer than continuum…

We construct isospectral pairs of Riemannian metrics on S^5 and on B^6, thus lowering by three the dimension of spheres and balls on which such metrics have been constructed previously (S^{n\ge 8} and B^{n\ge 9}). We also construct…

Certain semi-Riemannian metrics may be decomposed into a Riemannian part and an isochronal part. We use this idea and an idea of Kasner to construct a manifold in 6+1 Minkowski space with a well known metric. The full embedding we display…

This paper aims to provide an unsupervised modelling approach that allows for a more flexible representation of text embeddings. It jointly encodes the words and the paragraphs as individual matrices of arbitrary column dimension with unit…

Let $(M,d)$ be a bounded countable metric space and $c>0$ a constant, such that $d(x,y)+d(y,z)-d(x,z) \ge c$, for any pairwise distinct points $x,y,z$ of $M$. For such metric spaces we prove that they can be isometrically embedded into any…

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…