Abstract:
We review the relations between distance matrices and isometric embeddings and give simple proofs that distance matrices defined on euclidean and spherical spaces have all eigenvalues except one non-negative. Several generalizations are discussed.

Abstract:
An algebraic global isometric embedding of the nonrotating BTZ black hole is presented. The ambient spacetime is $\mathbb{M}^{2,3}$, the 3+2 dimensional flat spacetime. We also present the analogous embedding for the Euclidean BTZ spacetime and by performing a kind of double analytic continuation construct a 1-parameter family of embeddings of cosmological AdS spacetime into $\mathbb{M}^{2,3}$ which coincide asymptotically with the embedded BTZ manifold of the appropriate mass. Finally we note that the family of embeddings of cosmological AdS$_{n}$ into $\mathbb{M}^{2,n}$ generalises to higher dimensions.

Abstract:
We prove a rigidity theorem that shows that, under many circumstances, quasi-isometric embeddings of equal rank, higher rank symmetric spaces are close to isometric embeddings. We also produce some surprising examples of quasi-isometric embeddings of higher rank symmetric spaces. In particular, we produce embeddings of $SL(n,\mathbb R)$ into $Sp(2(n-1),\mathbb R)$ when no isometric embeddings exist.

Abstract:
Let $V$ be an $n$-dimensional left vector space over a division ring $R$. We write ${\mathcal G}_{k}(V)$ for the Grassmannian formed by $k$-dimensional subspaces of $V$ and denote by $\Gamma_{k}(V)$ the associated Grassmann graph. Let also $V'$ be an $n'$-dimensional left vector space over a division ring $R'$. Isometric embeddings of $\Gamma_{k}(V)$ in $\Gamma_{k'}(V')$ are classified in \cite{Pankov2}. A classification of $J(n,k)$-subsets in ${\mathcal G}_{k'}(V')$, i.e. the images of isometric embeddings of the Johnson graph $J(n,k)$ in $\Gamma_{k'}(V')$, is presented in \cite{Pankov1}. We characterize isometric embeddings of $\Gamma_{k}(V)$ in $\Gamma_{k'}(V')$ as mappings which transfer apartments of ${\mathcal G}_{k}(V)$ to $J(n,k)$-subsets of ${\mathcal G}_{k'}(V')$. This is a generalization of the earlier result concerning apartments preserving mappings \cite[Theorem 3.10]{Pankov-book}.

Abstract:
In this paper we consider piecewise linear (pl) isometric embeddings of Euclidean polyhedra into Euclidean space. A Euclidean polyhedron is just a metric space $\mathcal{P}$ which admits a triangulation $\mathcal{T}$ such that each $n$-dimensional simplex of $\mathcal{T}$ is affinely isometric to a simplex in $\mathbb{E}^n$. We prove that any 1-Lipschitz map from an $n$-dimensional Euclidean polyhedron $\mathcal{P}$ into $\mathbb{E}^{3n}$ is $\epsilon$-close to a pl isometric embedding for any $\epsilon > 0$. If we remove the condition that the map be pl then any 1-Lipschitz map into $\mathbb{E}^{2n + 1}$ can be approximated by a (continuous) isometric embedding. These results are extended to isometric embedding theorems of spherical and hyperbolic polyhedra into Euclidean space by the use of the Nash-Kuiper $C^1$ isometric embedding theorem. Finally, we discuss how these results extend to various other types of polyhedra.

Abstract:
An upper bound on the first S^1 invariant eigenvalue of the Laplacian for invariant metrics on the 2-sphere is used to find obstructions to the existence of isometric embeddings of such metrics in (R^3,can). As a corollary we prove: If the first four distinct eigenvalues have even multiplicities then the surface of revolution cannot be isometrically embedded in (R^3,can). This leads to a generalization of a classical result in the theory of surfaces.

Abstract:
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 embedding introduced in [BBG]. As t->0, asymptotic geometry of the embedded images is discussed.

Abstract:
This paper is devoted to the construction of norm-preserving maps between bounded cohomology groups. For a graph of groups with amenable edge groups we construct an isometric embedding of the direct sum of the bounded cohomology of the vertex groups in the bounded cohomology of the fundamental group of the graph of groups. With a similar technique we prove that if (X,Y) is a pair of CW-complexes and the fundamental group of each connected component of Y is amenable, the isomorphism between the relative bounded cohomology of (X,Y) and the bounded cohomology of X in degree at least 2 is isometric. As an application we provide easy and self-contained proofs of Gromov Equivalence Theorem and of the additivity of the simplicial volume with respect to gluings along \pi_1-injective boundary components with amenable fundamental group.

Abstract:
We investigate harmonic maps in the context of isometric embeddings when the target space is Ricci-flat and has codimension one. With the help of the Campbell-Magaard theorem we show that any $n$-dimensional ($n\geqslant 3$) Lorentzian manifold can be isometrically and harmonically embedded in a (n+1)-dimensional semi-Riemannian Ricci-flat space. We then extend our analysis to the case when the target space is an Einstein space. Finally, as an example, we work out the harmonic and isometric embedding of a Friedmann-Robertson-Walker spacetime in a five-dimensional Ricci-flat space and proceed to obtain a general scheme to minimally embed any vacuum solution of general relativity in Ricci-flat spaces with codimension one.

Abstract:
We prove some infinitesimal analogs of classical results of Menger, Schoenberg and Blumenthal giving the existence conditions for isometric embeddings of metric spaces in the finite-dimensional Euclidean spaces.