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The existence of isometric embedding of $S_q^m$ into $S_p^n$ , where $1\leq p\neq q\leq \infty$ and $m,n\geq 2$ , has been recently studied in [6]. In this article, we extend the study of isometric embeddability beyond the above-mentioned range of $p$ and $q$. More precisely, we show that there is no isometric embedding of the commutative quasi-Banach space $\ell _q^m(\mathbb {R})$ into $\ell _p^n(\mathbb {R})$ , where $(q,p)\in (0,\infty)\times (0,1)$ and $p\neq q$. As non-commutative quasi-Banach spaces, we show that there is no isometric embedding of $S_q^m$ into $S_p^n$ , where $(q,p)\in (0,2)\setminus \{1\}\times (0,1)$ $\cup \, \{1\}\times (0,1)\setminus \left \{\!\frac {1}{n}:n\in \mathbb {N}\right \}$ $\cup \, \{\infty \}\times (0,1)\setminus \left \{\!\frac {1}{n}:n\in \mathbb {N}\right \}$ and $p\neq q$. Moreover, in some restrictive cases, we also show that there is no isometric embedding of $S_q^m$ into $S_p^n$ , where $(q,p)\in [2, \infty)\times (0,1)$. A new tool in our paper is the non-commutative Clarkson's inequality for Schatten class operators. Other tools involved are the Kato–Rellich theorem and multiple operator integrals in perturbation theory, followed by intricate computations involving power-series analysis. [ABSTRACT FROM AUTHOR]

We study the case when a unit vector field ξ on a Riemannian manifold (M, g) defines an isometric embedding ξ : (M, g) Ñ (T1M, G˜) where G˜ is the Riemannian g-natural metric. The main goal is to find conditions under which the submanifold ξ(M) Ă (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) Ă (T1M, G˜). [ABSTRACT FROM AUTHOR]

In this short paper, we derive a new result on Umehara algebra. As a consequence, we prove that an indefinite complex hyperbolic space and an indefinite complex projective space do not share a common complex submanifold with induced metrics, answering a question raised in Cheng et al. [ABSTRACT FROM AUTHOR]

It was proved by S. Mazur and S. Ulam in 1932 that every isometric surjection between normed real vector spaces is affine. We generalize the Mazur–Ulam theorem and find necessary and sufficient conditions under which distance-preserving mappings between seminormed real vector spaces are linear. [ABSTRACT FROM AUTHOR]

We aim to generalize Figiel’s theorem to the local unit spheres case. Let T : S E ⟶ S F be an isometric embedding (not necessarily surjective) between unit spheres of Banach spaces. Does T admit the Figiel operator? Recently, some attempts were made by Liu and Yin (Acta Math Sci 43B(4):1503–1517, 2023), and a counterexample was given to this question, which shows that Figiel’s theorem cannot be routinely generalized to the unit spheres case. After that, a necessary condition (∗) cov { - T (C) ∪ T (- C) } ⊂ S F for every maximal convex subset C of S E was obtained for the existence of the Figiel operator. Naturally, Liu and Yin proposed the reformulated Figiel-type problem: is the existence of the Figiel operator for an isometric embedding T equivalent to the condition (∗) ? They answered this problem affirmatively in the case that E is a space with the Tingley property (T-property for short). To attack this problem, we introduce the concept of the slice approximating property (SAP for short) and show that all uniformly convex spaces, almost CL-spaces, and spaces with the T-property admit the SAP. Furthermore, we give an affirmative answer to the reformulated Figiel-type problem in the case that E is a local GL space with the SAP. This generalizes one of the main results in Liu and Yin (Acta Math Sci 43B(4):1503–1517, 2023) to a much more extensive case. At the end of this paper, an interesting result about isometric embedding and dimensions is proved. [ABSTRACT FROM AUTHOR]

A construction due to Burago and Zalgaller (Vestnik Leningrad Univ 15:66–80, 1960; St Petersburg Math J 7(3):369–385, 1995) shows that every orientable polyhedral surface, one that is obtained by gluing Euclidean polygons, has an isometric piecewise linear embedding into Euclidean space E 3 . A flat torus, resulting from the identification of the opposite sides of a Euclidean parallelogram, is a simple example of polyhedral surface. The embeddings constructed according to Burago and Zalgaller may have a huge number of vertices, moreover distinct for every flat torus. Based on another construction of Zalgaller (J Math Sci 100(3):2228–2238, 2000. https://doi.org/10.1007/s10958-000-0007-3) and on recent works by Arnoux et al. (2021, in preparation), we exhibit a universal triangulation with 2434 triangles which can be embedded linearly on each triangle in order to realize the metric of any flat torus. [ABSTRACT FROM AUTHOR]

Mixed data containing categorical and numerical attributes are widely available in real-world. Before analysing such data, it is typically necessary to process (transform/embed/represent) them into high-quality numerical data. The conditional probability transformation method (CPT) can provide acceptable performance in the majority of cases, but it is not satisfactory for datasets with strong attribute association. Inspired by the one dependence value difference metric method, the concept of relaxing the attributes conditional independence has been applied to CPT, but this approach has the drawback of dramatically-expanding the attribute dimensionality. We employ the isometric embedding method to tackle the problem of dimensionality expansion. In addition, an attribute weighting method based on the must-link and cannot-link constraints is designed to optimize the data transformation quality. Combining these methods, we propose an attribute-weighted isometric embedding (AWIE) for categorical encoding on mixed data. Extensive experimental results obtained on 16 datasets demonstrate that AWIE significantly improves upon the classification performance (increasing the F1-score by 2.54%, attaining 6/16 best results, and reaching average ranks of 1.94/8), compared with 28 competitors. [ABSTRACT FROM AUTHOR]

Two Kähler manifolds are called relatives if they admit a common Kähler submanifold with the same induced metrics. In this paper, we show that a Hartogs domain over an irreducible bounded symmetric domain equipped with the Bergman metric is not a relative to the complex Euclidean space. This generalizes the results in Cheng and Hao [On the non-existence of common submanifolds of Kähler manifolds and complex space forms. Ann Glob Anal Geom. 2021;60(1):167–180] and Cheng X, Niu Y. Submanifolds of Cartan–Hartogs domains and complex Euclidean spaces. J Math Anal Appl 2017;452(2):1262–1268.] and the novelty here is that the Bergman kernel of the Hartogs domain is not necessarily Nash algebraic. [ABSTRACT FROM AUTHOR]

A space of universal disposition is a Banach space which has certain natural extension properties for isometric embeddings of Banach spaces belonging to a specific class. We study spaces of universal disposition for non-archimedean Banach spaces. In particular, we introduce the classification of non-archimedean Banach spaces depending on the cardinality of maximal orthogonal sets, which can be viewed as a kind of special density and characterize spaces of universal disposition for each distinguished class. [ABSTRACT FROM AUTHOR]

We present a simple method to approximate the Fisher–Rao distance between multivariate normal distributions based on discretizing curves joining normal distributions and approximating the Fisher–Rao distances between successive nearby normal distributions on the curves by the square roots of their Jeffreys divergences. We consider experimentally the linear interpolation curves in the ordinary, natural, and expectation parameterizations of the normal distributions, and compare these curves with a curve derived from the Calvo and Oller's isometric embedding of the Fisher–Rao d-variate normal manifold into the cone of (d + 1) × (d + 1) symmetric positive–definite matrices. We report on our experiments and assess the quality of our approximation technique by comparing the numerical approximations with both lower and upper bounds. Finally, we present several information–geometric properties of Calvo and Oller's isometric embedding. [ABSTRACT FROM AUTHOR]

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 global isometric embeddings into 2 n -dimensional Euclidean space, of Hölder class C 1 , θ with θ < 1 / 3 for n = 2 and θ < (n + 2) − 1 for n ≥ 3. The proof is performed by Nash-Kuiper's convex integration construction and applying the gluing technique of the authors on short embeddings with small amplitude. [ABSTRACT FROM AUTHOR]

Here and hereafter, we identify HT ht with a 1-form (not relabelled). Thus HT ht is constant. Nomenclature Throughout HT ht is a simply-connected bounded domain with outward unit normal vectorfield HT ht . Therefore, by Eq. (3), HT ht equals zero. [Extracted from the article]

A graph is called a partial cube if it can be embedded into a hypercube isometrically. In this paper, we study a class of Cayley graphs—Cayley graphs generated by transpositions—and show that a Cayley graph Γ generated by transpositions is a partial cube if and only if Γ is a bubble sort graph. This result enhances a result of Alahmadi et al. in 2016: BSn is a partial cube. As a corollary, we give the analytical expressions of the Wiener indices of bubble sort graphs. [ABSTRACT FROM AUTHOR]

For a given metric measure space (X, d, μ) we consider finite samples of points, calculate the matrix of distances between them and then reconstruct the points in some finite-dimensional space using the multidimensional scaling (MDS) algorithm with this distance matrix as an input. We show that this procedure gives a natural limit as the number of points in the samples grows to infinity and the density of points approaches the measure μ. This limit can be viewed as "infinite MDS" embedding of the original space, now not anymore into a finite-dimensional space but rather into an infinitedimensional Hilbert space. We further show that this embedding is stable with respect to the natural convergence of metric measure spaces. However, contrary to what is usually believed in applications, we show that in many cases it does not preserve distances, nor is even bi-Lipschitz, but may provide snowflake (Assouad-type) embeddings of the original space to a Hilbert space (this is, for instance, the case of a sphere and a flat torus equipped with their geodesic distances). [ABSTRACT FROM AUTHOR]

We prove that every proper n -dimensional length metric space admits an "approximate isometric embedding" into Lorentzian space ℝ 3 n + 6 , 1 . By an "approximate isometric embedding" we mean an embedding which preserves the energy functional on a prescribed set of geodesics connecting a dense set of points. [ABSTRACT FROM AUTHOR]

This paper provides two new applications of conservation laws in biology. The first is the application of the van der Waals fluid formalism for action potentials. The second is the application of the conservation laws of differential geometry (Gauss–Codazzi equations) to produce non-smooth surfaces representing Endoplasmic Reticulum sheets. [ABSTRACT FROM AUTHOR]

We consider three different approaches to define natural Riemannian metricson polytopes of stochastic matrices. First, we define a natural class ofstochastic maps between these polytopes and give a metric characterization ofChentsov type in terms of invariance with respect to these maps. Second, weconsider the Fisher metric defined on arbitrary polytopes through theirembeddings as exponential families in the probability simplex. We show thatthese metrics can also be characterized by an invariance principle with respectto morphisms of exponential families. Third, we consider the Fisher metricresulting from embedding the polytope of stochastic matrices in a simplex ofjoint distributions by specifying a marginal distribution. All three approachesresult in slight variations of products of Fisher metrics. This is consistentwith the nature of polytopes of stochastic matrices, which are Cartesianproducts of probability simplices. The first approach yields a scaled productof Fisher metrics; the second, a product of Fisher metrics; and the third, aproduct of Fisher metrics scaled by the marginal distribution.

Two Kähler manifolds are called relatives if they admit a common Kähler submanifold with their induced metrics. In this paper, we provide a sufficient condition to determine whether a real analytic Kähler manifold is not a relative to a complex space form equipped with its canonical metric. As an application, we show that minimal domains, bounded homogeneous domains and some Hartogs domains equipped with their Bergman metrics are not relatives to the complex Euclidean space. [ABSTRACT FROM AUTHOR]

We study the possibility to explain the mystery of the dark matter (DM) through the transition from General Relativity to embedding gravity. This modification of gravity, which was proposed by Regge and Teitelboim, is based on a simple string-inspired geometrical principle: our spacetime is considered here as a four-dimensional surface in a flat bulk. We show that among the solutions of embedding gravity, there is a class of solutions equivalent to solutions of GR with an additional contribution of non-relativistic embedding matter, which can serve as cold DM. We prove the stability of such type of solutions and obtain an explicit form of the equations of motion of embedding matter in the non-relativistic limit. According to them, embedding matter turns out to have a certain self-interaction, which could be useful in the context of solving the core-cusp problem that appears in the Λ CDM model. [ABSTRACT FROM AUTHOR]

A subgraph H of a graph G is isometric if the distances between vertices in H coincide with the distances between the corresponding vertices in G. We show that for any integer n \geqslant 1, there is a graph on 3n+O(log2 n) vertices that contains isometric copies of all n-vertex graphs. Our main tool is a new type of distance labelling scheme, whose study might be of independent interest. [ABSTRACT FROM AUTHOR]

The notion of ultrametrics can be considered as a zero-dimensional analogue of ordinary metrics, and it is expected to prove ultrametric versions of theorems on metric spaces. In this paper, we provide ultrametric versions of the Arens-Eells isometric embedding theorem of metric spaces, the Hausdorff extension theorem of metrics, the Niemytzki-Tychonoff characterization theorem of the compactness, and the author's interpolation theorem of metrics and theorems on dense subsets of spaces of metrics. [ABSTRACT FROM AUTHOR]

A pseudo-edge graph of a convex polyhedron K is a 3-connected embedded graph in K whose vertices coincide with those of K, whose edges are distance minimizing geodesics, and whose faces are convex. We construct a convex polyhedron K in Euclidean 3-space with a pseudo-edge graph with respect to which K is not unfoldable. The proof is based on a result of Pogorelov on convex caps with prescribed curvature, and an unfoldability obstruction for almost flat convex caps due to Tarasov. Our example, which has 340 vertices, significantly simplifies an earlier construction by Tarasov, and confirms that Dürer's conjecture does not hold for pseudo-edge unfoldings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some topological properties of partial metric spaces (in short PMS). We give some relationship between metric-like PMS, sequentially isosceles PMS and sequentially equilateral PMS. We also prove a type of Urysohn's lemma for metric-like PMS. By applying the construction of Hartman–Mycielski, we show that every bounded PMS can be isometrically embedded into a pathwise connected and locally pathwise connected PMS. In the end, we show that a partial metric space is compact iff it is totally bounded and complete. [ABSTRACT FROM AUTHOR]

Data are often represented as graphs. Many common tasks in data science are based on distances between entities. While some data science methodologies natively take graphs as their input, there are many more that take their input in vectorial form. In this survey, we discuss the fundamental problem of mapping graphs to vectors, and its relation with mathematical programming. We discuss applications, solution methods, dimensional reduction techniques, and some of their limits. We then present an application of some of these ideas to neural networks, showing that distance geometry techniques can give competitive performance with respect to more traditional graph-to-vector mappings. [ABSTRACT FROM AUTHOR]

The isometric Ramsey numberIR(H→) of a family H→ of digraphs is the smallest number of vertices in a graph G such that any orientation of the edges of G contains every member of H→ in the distance‐preserving way. We observe that the isometric Ramsey number of a finite family of finite acyclic digraphs is always finite, and present some bounds in special cases. For example, we show that the isometric Ramsey number of the family of all oriented trees with n vertices is at most n2n+o(n). [ABSTRACT FROM AUTHOR]

This paper revisits the Gneiting class of positive definite kernels originally proposed as a class of covariance functions for space-time processes. Under the framework of quasi-metric spaces and isometric embeddings, the paper proposes a general and unifying framework that encompasses results provided by earlier literature. Our results allow to study the positive definiteness of the Gneiting class over products of either Euclidean spaces or high dimensional spheres and quasi-metric spaces. In turn, Gneiting's theorem is proved here by a direct construction, eluding Fourier inversion (the so-called Gneiting's lemma) and convergence arguments that are required by Gneiting to preserve an integrability assumption. [ABSTRACT FROM AUTHOR]

We study the procedure that allows to rewrite the actions of some modified gravity theories like mimetic gravity and Regge–Teitelboim embedding theory as Einstein–Hilbert actions with additional matter contributions. It turns out that this procedure can be applied to brane action, which leads to a polynomial brane action. We also examine the connection between this procedure and Polyakov trick in string theory. [ABSTRACT FROM AUTHOR]

A Boolean algebra A equipped with a (finitely-additive) positive probability measure m can be turned into a metric space (A , d m) , where d m (a , b) = m ((a ∧ ¬ b) ∨ (¬ a ∧ b)) , for any a , b ∈ A , sometimes referred to as metric Boolean algebra. In this paper, we study under which conditions the space of atoms of a finite metric Boolean algebra can be isometrically embedded in R N (for a certain N) equipped with the Euclidean metric. In particular, we characterize the topology of the positive measures over a finite algebra A such that the metric space (At (A) , d m) embeds isometrically in R N (with the Euclidean metric). [ABSTRACT FROM AUTHOR]

A semi-global isometric embedding of abstract surfaces with Gaussian curvature changing signs of any finite order is obtained through solving the Darboux equation. [ABSTRACT FROM AUTHOR]

Let Λ∗ be the free monoid of (finite) words over a not necessarily finite alphabet Λ, which is equipped with some (partial) order. This ordering lifts to Λ∗, where it extends the divisibility ordering of words. The MacNeille completion of Λ∗ constitutes a complete lattice ordered monoid and is realized by the system of "closed" lower sets in Λ∗ (ordered by inclusion) or its isomorphic copy formed of the "closed" upper sets (ordered by reverse inclusion). Under some additional hypothesis on Λ, one can easily identify the closed lower sets as the finitely generated ones, whereas it is more complicated to determine the closed upper sets. For a fairly large class of ordered sets Λ (including complete lattices as well as antichains) one can generate the closure of any upper set of words by means of binary operations ("syntactic rules") thus obtaining an efficient procedure to test closedness. Closed upper sets of words are involved in an embedding theorem for valuated oriented graphs. In fact, generalized paths (so-called "zigzags") are encoded by words over an alphabet Λ. Then the valuated oriented graphs which are "isometrically" embeddable in a product of zigzags have the characteristic property that the words corresponding to the zigzags between any pair of vertices form a closed upper set in Λ∗. [ABSTRACT FROM AUTHOR]

Abstract It is known that if finite subsets of a locally finite metric space M admit C -bilipschitz embeddings into ℓ p (1 ≤ p ≤ ∞) , then for every ε > 0 , the space M admits a (C + ε) -bilipschitz embedding into ℓ p. The goal of this paper is to show that for p ≠ 2 , ∞ this result is sharp in the sense that ε cannot be dropped out of its statement. [ABSTRACT FROM AUTHOR]

We study the regularity of the isometric embedding X : (B(O,r),g) → (R3,gcan) of a 2-ball with nonnegatively curved C4 metric into R3. Under the assumption that X can be expressed in the graph form, we show X ∈ C2,1 near P, which is optimal by Iaia's example. [ABSTRACT FROM AUTHOR]

For an evolution of metrics (M , g t) there is a t-smooth family of embeddings e t : M → R N inducing g t , but in general there is no family of embeddings extending a given initial embedding e 0 . We give an example of this phenomenon when g t is the evolution of g 0 under the Ricci flow. We show that there are embeddings e 0 inducing g 0 which do not admit of t-smooth extensions to e t inducing g t for any t > 0 . We also find hypersurfaces of dim > 2 that will not remain a hypersurface under Ricci flow for any positive time. [ABSTRACT FROM AUTHOR]

Abstract Motivated by the quasi-local mass problem in general relativity, we study the rigidity of isometric immersions with the same mean curvature into a warped product space. As a corollary of our main result, two star-shaped hypersurfaces in a spatial Schwarzschild or AdS-Schwarzschild manifold with nonzero mass differ only by a rotation if they are isometric and have the same mean curvature. We also prove similar results if the mean curvature condition is replaced by an σ 2 -curvature condition. [ABSTRACT FROM AUTHOR]

Urban environments are restricted by various physical, regulatory and customary barriers such as buildings, one-way systems and pedestrian crossings. These features create challenges for predictive modelling in urban space, as most proximity-based models rely on Euclidean (straight line) distance metrics which, given restrictions within the urban landscape, do not fully capture spatial urban processes. Here, we argue that road distance and travel time provide effective alternatives, and we develop a new low-dimensional Euclidean distance metric based on these distances using an isomap approach. The purpose of this is to produce a valid covariance matrix for Kriging. Our primary methodological contribution is the derivation of two symmetric dissimilarity matrices ( and ), with which it is possible to compute low-dimensional Euclidean metrics for the production of a positive definite covariance matrix with commonly utilised kernels. This new method is implemented into a Kriging predictor to estimate house prices on 3,669 properties in Coventry, UK. We find that a  metric estimating a combination of road distance and travel time, in both and , produces a superior house price predictor compared with alternative state-of-the-art methods, that is, a standard Euclidean metric in and a non-restricted road distance metric in  and . F [ABSTRACT FROM AUTHOR]