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This paper is concerned with the existence theory of isometric immersions of surfaces with negative Gaussian curvature into the 3-dimensional Euclidean space. We reformulate the Gauss--Codazzi equations, \emph{i.e.}, the partial differential equations for isometric immersions, into hyperbolic conservation laws for the flows of Chaplygin gas with nonzero source terms. Then, by employing the theories of invariant regions and compensated compactness, we establish the existence of $W^{2,p}$-isometric immersions for several general families of metrics, with any finite index $p$ and over arbitrarily large infinite strips or rectangular domains. Such metrics include those of various classical minimal surfaces: helicoid, catenoid, pseudosphere, and Enneper surfaces, as well as metrics in isothermal coordinates or of the ``reciprocal-type''. In our fluid dynamical formulation of the isometric immersion problem, we specialise in the case that the two Riemann invariants for the associated hyperbolic conservation law remain bounded and of distinctive signs, and obtain $L^p$-solutions to the initial-boundary value problem via entropy analysis. The isometric immersions constructed in this paper

This paper is devoted to investigating the isometric immersion problem of Riemannian manifolds in a high codimension. It has recently been demonstrated that any short immersion from an $n$-dimensional smooth compact manifold into $2n$-dimensional Euclidean space can be uniformly approximated by $C^{1,θ}$ isometric immersions with any $θ\in(0,1/(n+2))$ in dimensions $n\geq3$. In this paper, we improve the Hölder regularity of the constructed isometric immersions in the local setting, achieving $C^{1,θ}$ for all $θ\in(0,1/n)$ in odd dimensions and all $θ\in(0,1/(n+1))$ in even dimensions. Moreover, we also establish explicit $C^{1}$ estimates for the isometric immersions, which indicate that the larger the initial metric error is, the greater the $C^{1}$ norms of the resulting isometric maps become, meaning that their slope become steeper.

A set $S$ of isometric paths of a graph $G$ is ``$v$-rooted'', where $v$ is a vertex of $G$, if $v$ is one of the endpoints of all the isometric paths in $S$. The isometric path complexity of a graph $G$, denoted by $ipco{G}$, is the minimum integer $k$ such that there exists a vertex $v\in V(G)$ satisfying the following property: the vertices of any single isometric path $P$ of $G$ can be covered by $k$ many $v$-rooted isometric paths. First, we provide an $O(n^2 m)$-time algorithm to compute the isometric path complexity of a graph with $n$ vertices and $m$ edges. Then we show that the isometric path complexity remains bounded for graphs in three seemingly unrelated graph classes, namely, hyperbolic graphs, (theta, prism, pyramid)-free graphs, and outerstring graphs. There is a direct algorithmic consequence of having small isometric path complexity. Specifically, we show that if the isometric path complexity of a graph $G$ is bounded by a constant, then there exists a polynomial-time constant-factor approximation algorithm for ISOMETRIC PATH COVER, whose objective is to cover all vertices of a graph with a minimum number of isometric paths. This applies to all the above graph cl

An \textit{isometric path} is a shortest path between two vertices. An \textit{isometric path partition} (IPP) of a graph $G$ is a set $\mathcal{I}$ of vertex-disjoint isometric paths in $G$ that partition the vertices of $G$. The \textit{isometric path partition number} of $G$, denoted by $\text{ipp}(G)$, is the minimum cardinality of an IPP of~$G$. An \textit{induced path partition} (IndPP) of a graph $G$ is a set $\mathcal{I}$ of vertex-disjoint induced paths in~$G$ that partition the vertices of $G$. The \textit{induced path partition number} of $G$, denoted by $\text{indpp}(G)$, is the minimum cardinality of an IndPP of $G$. In this article, we study both these parameters and observe that every graph $G$ satisfies $\text{indpp}(G) \leq \text{ipp}(G) \leq |V(G)| - ν(G)$, where $ν(G)$ is the matching number of $G$. We further prove that a connected graph $G$ is extremal with respect to this upper bound, i.e.\ satisfies $\text{ipp}(G) = |V(G)| - ν(G)$, (resp.\ $\text{indpp}(G) = |V(G)| - ν(G)$), if and only if either (i) all blocks of $G$ are odd complete graphs, or (ii) all blocks of $G$ except one are odd complete graphs, and the unique block $B$ of $G$ that is not an odd compl

This note pertains to isometric embeddings endowed with certain geometric properties. We study two embedding problems for a Riemannian manifold $M$ which is diffeomorphic to $\RR^n$ and admits a Bieberbach group $Γ$ acting by isometries. The first problem concerns the existence of an isometric embedding of $M$ into a bounded subset of some Euclidean space $\RR^{D_1}$. The second problem seeks a $Γ$-equivariant isometric embdding of $M$ into $\RR^{D_2}$. By using a known trick in a novel way, our idea yields results with $D_1 = N+2n$ and $D_2 = N+n$, where $N$ is the Nash dimension of $ M/Γ$. Moreover, we also show that an $n$-dimensional smooth manifold, of Nash dimension $N$, can be isometrically embedded into a bounded subset of $\RR^{2N}$.

Pillow boxes are surfaces used for gift boxes, packaging, and even architectural applications. By definition, a pillow box is isometric to a double rectangle consisting of two copies of a rectangle. If the crease pattern is allowed to change, there exist continuous isometric deformations from a pillow box to a double rectangle. However, practical applications often require preserving the crease pattern. In this paper, we classify isometric deformations from a pillow box to a double rectangle among curved foldings that preserve the crease pattern. As a corollary, we prove that such an isometric deformation necessarily changes the topology of a pillow box.

Isometric class of minimal surfaces in the Euclidean 3-space $\mathbb{R}^3$ has the rigidity: if two simply connected minimal surfaces are isometric, then one of them is congruent to a surface in the specific one-parameter family, called the associated family, of the other. On the other hand, the situation for surfaces with Lorentzian metrics is different. In this paper, we show that there exist two timelike minimal surfaces in the Lorentz-Minkowski 3-space $\mathbb{R}^3_1$ that are isometric each other but one of which does not belong to the congruent class of the associated family of the other. We also prove a rigidity theorem for isometric and anti-isometric classes of timelike minimal surfaces under the assumption that surfaces have no flat points. Moreover, we show how symmetries of such surfaces propagate for various deformations including isometric and anti-isometric deformations. In particular, some conservation laws of symmetry for Goursat transformations are discussed.

We show that any isometric action of a residually finite group admits approximate local finite models. As a consequence, if $G$ is residually finite, every isometric $G$-action embeds isometrically into a metric ultraproduct of finite isometric $G$-actions.

The classical Banach--Mazur theorem asserts that every separable Banach space admits an isometric embedding into $C[0,1]$. It is also well known that every separable Banach space embeds isometrically into $\ell^\infty$. We show that such an embedding can be chosen so that its image intersects $c$ only at the origin. Moreover, we prove that any finite- or countable-dimensional, or more generally separable, subspace of $(\ell^\infty \setminus c)\cup\{0\}$ can be extended to a subspace containing an isometric copy of an arbitrary separable Banach space, while still avoiding $c$. We further establish that this extension property also holds for every subspace $D\subset \ell^\infty$ with $D\cap c=\{0\}$ and separable image in the quotient $\ell^\infty/c$.

We study the problem of existence and uniqueness of isometric Banach preduals of a Banach space. We derive necessary and sufficient conditions for the existence of an isometric Banach predual of a Banach space $X$. Then we focus on the case that $X=\mathcal{F}(Ω)$ is a Banach space of scalar-valued functions on a non-empty set $Ω$ and describe those spaces which admit a special isometric Banach predual, namely a \emph{strong isometric Banach linearisation}, i.e. there is a Banach space $Y$, a map $δ\colonΩ\to Y$ and an isometric isomorphism $T\colon\mathcal{F}(Ω)\to Y^{\ast}$ such that $T(f)\circ δ= f$ for all $f\in\mathcal{F}(Ω)$. Finally, we give necessary and sufficient conditions for Banach spaces $\mathcal{F}(Ω)$ with a strong isometric Banach linearisation to have a (strongly) unique isometric Banach predual.

The isometric path cover (partition) problem of a graph is to find a minimum set of isometric paths which cover (partition) the vertex set of the graph. The isometric path cover (partition) number of a graph is the cardinality a minimum isometric path cover (partition). We prove that the isometric path partition problem and the isometric $k$-path partition problem for $k\geq 3$ are NP-complete on general graphs. Fisher and Fitzpatrick \cite{FiFi01} have shown that the isometric path cover number of $(r\times r)$-dimensional grid is $\lceil 2r/3\rceil$. We show that the isometric path cover (partition) number of $(r\times s)$-dimensional grid is $s$ when $r \geq s(s-1)$. We establish that the isometric path cover (partition) number of $(r\times r)$-dimensional torus is $r$ when $r$ is even and is either $r$ or $r+1$ when $r$ is odd. Then, we demonstrate that the isometric path cover (partition) number of an $r$-dimensional Benes network is $2^r$. In addition, we provide partial solutions for the isometric path cover (partition) problems for cylinder and multi-dimensional grids.

We investigate isometric immersions $f\colon M^n\to\R^{n+2}$, $n\geq 3$, of Riemannian manifolds into Euclidean space with codimension two that admit isometric deformations that preserve the metric of the Gauss map. In precise terms, the preservation of the third fundamental form of the submanifold must be ensured throughout the deformation. For minimal isometric deformations of minimal submanifolds this is always the case. Our main result is of a local nature and states that if $f$ is neither minimal nor reducible, then it is a hypersurface of an isometrically deformable hypersurface $F\colon\tilde{M}^{n+1}\to\R^{n+2}$ such that the deformations of $F$ induce those of $f$. Moreover, for a particular class of such submanifolds, a complete local parametric description is provided.

Isometric covariant representations play an important role in the study of Cuntz-Pimsner algebras. In this article, we study partial isometric covariant representations and explore under what conditions powers and roots of partial isometric covariant representations are also partial isometric covariant representations.

Let $G$ and $H$ be locally compact groups. We will show that each contractive Jordan isomorphism $Φ\colon L^1(G)\to L^1(H)$ is either an isometric isomorphism or an isometric anti-isomorphism. We will apply this result to study isometric two-sided zero product preservers on group algebras and, further, to study local and approximately local isometric automorphisms of group algebras.

An edit distance is a metric between words that quantifies how two words differ by counting the number of edit operations needed to transform one word into the other one. A word f is said isometric with respect to an edit distance if, for any pair of f-free words u and v, there exists a transformation of minimal length from u to v via the related edit operations such that all the intermediate words are also f-free. The adjective 'isometric' comes from the fact that, if the Hamming distance is considered (i.e., only mismatches), then isometric words are connected with definitions of isometric subgraphs of hypercubes. We consider the case of edit distance with swap and mismatch. We compare it with the case of mismatch only and prove some properties of isometric words that are related to particular features of their overlaps.

In this paper we construct a closed subspace $X\subset C[0,1]$ with countable oscillating spectrum $Ω(X)$ such that $X$ is isometric to $\ell^1$. This provides a negative answer to Question~4.3 posed by Enflo, Gurariy, and Seoane in [Trans. Amer. Math. Soc. \textbf{366} (2014)].

We study probability-measure preserving (p.m.p.) actions of finitely generated groups via the graphings they define. We introduce and study the notion of isometric orbit equivalence for p.m.p. actions: two p.m.p. actions are isometric orbit equivalent if the graphings defined by some fixed generating systems of the groups are measurably isometric. We highlight two kind of phenomena. First, we prove that the notion of isometric orbit equivalence is rigid for groups whose Cayley graph, with respect to a fixed generating system, has a countable group of automorphism. On the other hand, we introduce a general construction of isometric orbit equivalent p.m.p. actions, which leads to interesting nontrivial examples of isometric orbit equivalent p.m.p. actions for the free group. In particular, our examples show that mixing is not invariant under isometric orbit equivalence.

Pulling back complex structures along a branched covering induces a holomorphic isometric embedding of Teichmüller spaces. We show that for dimension at least $2$, all isometric embeddings arise from branched coverings. This generalizes a theorem of Royden. As a consequence we obtain that totally geodesic submanifolds of Teichmüller space, which are isometric to some Teichmüller space, are covering constructions. Another consequence is the classification of locally isometric embeddings of moduli spaces of Riemann surfaces.

We investigate the notion of subsystem in the framework of spectral triple as a generalized notion of noncommutative submanifold. In the case of manifolds, we consider several conditions on Dirac operators which turn embedded submanifolds into isometric submanifolds. We then suggest a definition of spectral subtriple based on the notion of submanifold algebra and the already existing notions of Riemannian, isometric, and totally geodesic morphisms. We have shown that our definitions work at least in some relevant almost commutative examples.