Quantum gravity is expected to impose constraints on the moduli spaces of massless fields that can arise in effective quantum field theories. A recent proposal asserts that the asymptotic volume growth of these spaces is severely restricted, and related to the existence of duality symmetries. In this work we link this proposal to a tameness criterion, by suggesting that any consistent moduli space should admit a tame isometric embedding into Euclidean space. This allows us to promote the volume growth constraint to a local condition, and give the growth coefficient a geometric interpretation in terms of complexity. We study the implications of this proposal for the emergence of dualities, as well as for the curvature and infinite distance limits of moduli spaces.

We prove that a generic reservoir system admits a generalized synchronization that is a topological embedding of the input system's attractor. We also prove that for sufficiently high reservoir dimension (given by Nash's embedding theorem), there exists an isometric embedding generalized synchronization. The isometric embedding can be constructed explicitly when the reservoir system and source dynamics are linear.

Abstract An embedded twisted paper cylinder of aspect ratio λ is a smooth isometric embedding of a flat λ × 1 cylinder into ℝ 3 such that the images of the boundary components are linked. We prove that for such an object to exist we must have λ > 2 and that this bound is sharp. We also show that any sequence of examples having aspect ratio converging to 2 must converge to a (non-smooth) 4-fold wrapping of a right-angled isosceles triangle.

Abstract Quantum gravity in 4D asymptotically flat spacetimes features spontaneous symmetry breaking due to soft radiation hair, intimately tied to the proliferation of IR divergences. A holographic description via a putative 2D CFT is expected free of such redundancies. In this series of two papers, we address this issue by initiating the study of quantum error correction in celestial CFT (CCFT). In part I we construct a toy model with finite degrees of freedom by revisiting noncommutative geometry in Kleinian hyperkähler spacetimes. The model obeys a Wick algebra that renormalizes in the radial direction and admits an isometric embedding á la Gottesman–Kitaev–Preskill. The code subspace is composed of two-qubit stabilizer states which are robust under soft spacetime fluctuations. Symmetries of the hyperkähler space become discrete and translate into the Clifford group familiar from quantum computation. The construction is then embedded into the incidence relation of twistor space, paving the way for the CCFT regime addressed in follow-up work.

We introduce new flatness coefficients, which we shall call \iota -numbers , for Ahlfors k -regular sets in metric spaces ( k\in \mathbb{N} ). Using these coefficients for k=1 , we characterize uniform 1 -rectifiability in rather general metric spaces, completing earlier work by Hahlomaa and Schul. Our proof proceeds by quantifying an isometric embedding theorem due to Menger, and by an abstract argument that allows to pass from a local covering by continua to a global covering by 1 -regular connected sets.

Our purpose in this paper is to study isometries and isometric embeddings of the p -Wasserstein space \mathcal{W}_{p}(\mathbb{H}^{n}) over the Heisenberg group \mathbb{H}^{n} for all p>1 and for all n\geq1 . First, we create a link between optimal transport maps in the Euclidean space \mathbb{R}^{2n} and the Heisenberg group \mathbb{H}^{n} . Then we use this link to understand isometric embeddings of \mathbb{R} and \mathbb{R}_{+} into \mathcal{W}_{p}(\mathbb{H}^{n}) for p>1 . That is, we characterize complete geodesics and geodesic rays in the Wasserstein space. Using these results, we determine the metric rank of \mathcal{W}_{p}(\mathbb{H}^{n}) . Namely, we show that \mathbb{R}^{k} can be embedded isometrically into \mathcal{W}_{p}(\mathbb{H}^{n}) for p>1 if and only if k\leq n . As a consequence, we conclude that \mathcal{W}_{p}(\mathbb{R}^{k}) and \mathcal{W}_{p}(\mathbb{H}^{k}) can be embedded isometrically into \mathcal{W}_{p}(\mathbb{H}^{n}) if and only if k\leq n . In the second part of the paper, we study the isometry group of \mathcal{W}_{p}(\mathbb{H}^{n}) for p>1 . We find that these spaces are all isometrically rigid, meaning that for every isometry \Phi\colon \mathcal{W}_{p}(\mathbb{H}^{n})\to\mathcal{W}_{p}(\mathbb{H}^{n}) , there exists an isometry \psi\colon\mathbb{H}^{n}\to\mathbb{H}^{n} such that \Phi=\psi_{\#} .

Abstract Lying at the intersection of Ado's theorem and the Nash embedding theorem, we consider the problem of finding faithful representations of Lie groups that are simultaneously isometric embeddings. Such special maps are found for a certain class of solvable Lie groups that includes all Einstein and Ricci soliton solvmanifolds, as well as all Riemannian 2‐step nilpotent Lie groups. As a consequence, we extend work of Tamaru by showing that all Einstein solvmanifolds can be realized as submanifolds (in the submanifold geometry) of a symmetric space.

Isometric embeddings of Teichmüller spaces don't extend to the Gardiner-Masur compactification

Let \(G\) be a connected graph and let \(d_G\) be the geodesic distance on \(V(G)\). The metric spaces \((V(G), d_{G})\) were characterized up to isometry for all finite connected \(G\) by David C. Kay and Gary Chartrand in 1965. The main result of this paper expands this characterization on infinite connected graphs. We also prove that every metric space with integer distances between its points admits an isometric embedding in \((V(G), d_G)\) for suitable \(G\).

We propose a deterministic method to find all holographic entropy inequalities that have corresponding contraction maps and argue the completeness of our method. We use a triality between holographic entropy inequalities, contraction maps and partial cubes. More specifically, the validity of a holographic entropy inequality is implied by the existence of a contraction map, which we prove to be equivalent to finding an isometric embedding of a contracted graph. Thus, by virtue of the argued completeness of the contraction map proof method, the problem of finding all holographic entropy inequalities is equivalent to the problem of finding all contraction maps, which we translate to a problem of finding all image graph partial cubes. We give an algorithmic solution to this problem and characterize the complexity of our method. We also demonstrate interesting by-products, most notably, a procedure to generate candidate quantum entropy inequalities.

A classic result of Shi and Tam states that a 2-sphere of positive Gauss and mean curvature bounding a compact 3-manifold with nonnegative scalar curvature must have total mean curvature not greater than that of the isometric embedding into Euclidean 3-space, with equality only for domains in this reference manifold. We generalize this result to 2-tori of Gauss curvature greater than − 1 -1 , which bound a compact 3-manifold having scalar curvature not less than − 6 -6 and at least one other boundary component satisfying a ‘trapping condition’. The conclusion is that the total weighted mean curvature is not greater than that of an isometric embedding into the Kottler manifold, with equality only for domains in this space. Examples are given to show that the assumption of a secondary boundary component cannot be removed. The result gives a positive mass theorem for the static Brown-York mass of tori, in analogy to the Shi-Tam positivity of the standard Brown-York mass, and represents the first such quasi-local mass positivity result for nonspherical surfaces. Furthermore, we prove a Penrose-type inequality in this setting.

The object of this paper is to find a vector field ξ and a constant λ on an n-dimensional compact Riemannian manifold Mn,g such that we obtain the Ricci soliton Mn,g,ξ,λ. In order to achieve this objective, we choose an isometric embedding provided in the work of Kuiper and Nash in the Euclidean space Rm,g¯ and choose ξ as the tangential component of a constant unit vector on Rm and call it a Kuiper–Nash vector. If τ is the scalar curvature of the compact Riemannian manifold Mn,g with a Kuiper–Nash vector ξ, we show that if the integral of the function ξτ has a suitable lower bound containing a constant λ, then Mn,g,ξ,λ is a Ricci soliton; we call this a Kuiper–Nash Ricci soliton. We find a necessary and sufficient condition involving the scalar curvature τ under which a compact Kuiper–Nash Ricci soliton Mn,g,ξ,λ is a trivial soliton. Finally, we find a characterization of an n-dimensional compact trivial Kuiper–Nash Ricci soliton Mn,g,ξ,λ using an upper bound on the integral of divξ2 containing the scalar curvature τ.

Abstract We prove a codimension reduction and congruence theorem for compact ‐dimensional submanifolds of that admit a mean convex isometric embedding into using a Reilly‐type formula for space forms.

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 for stable commutator length. More generally, we show that if T is a subsurface of an oriented compact (possibly closed) connected surface S , and c is an integral 1 -chain on \pi_{1}T , then there is an isometric embedding H_{2}(T,c)\to H_{2}(S,c) for the relative Gromov seminorm. Those statements are proved by finding an appropriate standard form for admissible surfaces and showing that, under the right homology vanishing conditions, such an admissible surface in S for a chain in T is in fact an admissible surface in T .

Abstract We prove that for any Riemannian metric $g$ on a closed orientable surface $\Sigma $ and any spacelike embedding $f:\Sigma \rightarrow M$ in a pseudo-Riemannian manifold $(M,h)$, the embedding $f$ can be $C^{0}$-approximated by a smooth conformal embedding for $g$. If in addition, $M$ is a quotient of the $(2+1)$-dimensional solid timelike cone by a cocompact lattice of $SO^{\circ }(2,1)$, we show that the set of negatively curved metrics on $\Sigma $ that admit isometric embeddings in $M$ projects into a relatively compact set in the Teichmüller space.

Recently, A. Zaitov suggested a new metric on the space of all nonempty compact subsets of a given metric space. In the present paper we show that this metric turns the hyperspace functor exp into a perfect metrizable functor. Further, we establish that the hyperspace functor has many remarkable properties with respect to this metrization. In particular, this functor preserves isometric embeddings.

Conformal invariants of isometric embeddings of the smooth metrics of a surface

Holomorphic isometric embeddings of complex Grassmannians into quadrics: The general case

We study the case when a unit vector field ξ on a Riemannian manifold (M,g) defines an isometric embedding ξ:(M,g)→(T1M, G), where G is the Riemannian g-natural metric. The main goal is to find conditions under which the submanifold ξ(M)⊂(T1M, G) can be totally geodesic. It is proved that the Reeb vector field of a K-contact metric structure on M gives rise to totally geodesic ξ(M) if and only if the structure is Sasakian. As a by-product, we find the expression for the second fundamental form of ξ(M)⊂(T1M, G).

On the isometric version of Whitney's strong embedding theorem