Isometric embeddings of Teichmüller spaces are derived from branched coverings.
problem Characterizing isometric embeddings of Teichmüller spaces.
method Holomorphic isometric embeddings induced by branched coverings.
result All isometric embeddings of Teichmüller spaces of dimension at least 2 arise from branched coverings.
Heat kernels map RCD spaces to Riemannian manifolds.
problem Mapping RCD spaces to Riemannian manifolds using heat kernels.
method Using heat kernels to map RCD spaces into L2 space and then normalizing to achieve isometric immersions. result Compact RCD spaces with isometrically heat kernel immersions are isometric to unweighted smooth Riemannian manifolds.
The paper examines rigidity of metric constructions in Wasserstein spaces.
problem Isometric rigidity of metric constructions in Wasserstein spaces.
method Analyzes spaces like Hilbert, rays, half-cylinders, and spherical suspensions.
result Different spaces exhibit varying levels of isometric rigidity in Wasserstein spaces.
The paper examines isometric timelike surfaces in 4D Minkowski space.
problem Analyzing geometric properties of isometric timelike surfaces.
method Study of Bour's theorem for four kinds of timelike helicoidal surfaces, analysis of geometric properties, presentation of parametrizations.
result Introduction of isometric pairs of timelike surfaces with same Gauss map.
We study quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semisimple higher rank Lie groups. We show that any quasi-isometric embedding between symmetric spaces of the same rank can be decomposed into a product of quasi-isometric embeddings into irreducible symmetric spaces. We thu…
Paper finds isometric timelike minimal surfaces with unique properties.
problem Rigidity of isometric timelike minimal surfaces in Lorentz-Minkowski space.
method Analyzes symmetries and deformations of timelike minimal surfaces.
result Existence of isometric timelike minimal surfaces not congruent to associated family.
Proves isometric embeddings in Euclidean spaces for RCD spaces.
problem Isometric immersions of RCD spaces in Euclidean spaces.
method Analyzes regular isometric immersions and eigenmaps of compact non-collapsed RCD spaces.
result Eigenmaps of compact non-collapsed RCD spaces are locally bi-Lipschitz embeddings to spheres.
We study isometric cohomogeneity one actions on the (n+1)-dimensional Minkowski space up to orbit-equivalence. We give examples of isometric cohomogeneity one actions on the Minkowski space whose orbit spaces are non-Hausdorff. We show that there exist isometric cohomogeneity one actions on the Minkowski space which ar…
Study minimal Kähler submanifolds in product of space forms.
problem Existence and properties of minimal isometric immersions of Kähler manifolds into product spaces.
method Analyse obstruction conditions and prove existence results for minimal immersions into Sm−1imesR and Hm−1imesR. result Only minimal isometric immersions of Riemannian surfaces into Sm−1imesR and Hm−1imesR exist. First explicit isometric immersion of a flat Klein bottle in 3D space.
problem Finding an isometric embedding of a Klein bottle in 3D.
method Piecewise-linear map from a Klein bottle to Euclidean 3-space.
result Explicit numerical data for a flat Klein bottle isometrically immersed in 3D.
Rigidity of Wasserstein spaces over Riemannian manifolds
problem Isometric rigidity of L2 Wasserstein spaces over Riemannian manifolds
method Showing L2 Wasserstein spaces are isometrically rigid if and only if their underlying manifolds do not admit a Euclidean de Rham factor
result Isometry of L2 Wasserstein spaces over non-Euclidean manifolds
Proves symmetric spaces for certain quasi-isometric properties.
problem Understanding quasi-isometric spaces and their properties.
method Analyzes geometric and geometrically finite quasi-actions.
result Proves quasi-isometric spaces are symmetric or real line.
No isometric immersion of hyperbolic space into Euclidean space.
problem Isometric immersions of hyperbolic space into Euclidean space.
method Generalized Gauss-Bonnet formula for Riemannian polyhedra.
result Hilbert's Theorem generalized to higher dimensions.
Study holomorphic isometric embeddings of a Grassmannian into quadrics.
problem Holomorphic isometric embeddings of complex Grassmannian into quadrics.
method Generalization of do Carmo--Wallach theory to study moduli space.
result Moduli space of embeddings up to equivalence discussed.
Due to Janet-Cartan's theorem, any analytic Riemannian manifolds can be locally isometrically embedded into a sufficiently high dimensional Euclidean space. However, for an individual Riemannian manifold (M,g), it is in general hard to determine the least dimensional Euclidean space into which (M,g) can be locally isom…
We prove that every proper n-dimensional length metric space admits an "approximate isometric embedding" into Lorentzian space R3n+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.
Paper shows non-CSC HCMU metrics can't be isometrically immersed into 3D space forms.
problem Non-CSC HCMU metrics cannot be isometrically immersed into 3D space forms.
method Using moving frames to demonstrate the impossibility of isometric minimal immersion.
result Non-CSC HCMU metrics cannot be isometrically immersed into 3D space forms.
Stability result for nearly isometric subspaces and Finsler surfaces.
problem Stability of normed spaces and Finsler surfaces under near-isometric conditions.
method Refined topological argument and explicit quantification using Banach-Mazur distance.
result A 2-dimensional surface with near-monochromatic Finsler metric is approximately Riemannian.
New criteria for non-isometric group actions in metric spaces.
problem Understanding group actions on non-isometric spaces.
method Generalizing results from isometric to continuous group actions.
result Criterion for cocompact cyclic groups to be inessential.
The paper explores almost paracomplex structures on 4-manifolds and their properties.
problem Existence and properties of isometric and anti-isometric almost paracomplex structures on pseudo-Riemannian 4-manifolds.
method Analyzes the conditions under which these structures are parallel and their implications for the scalar curvature and Einstein metrics.
result If an isometric or anti-isometric almost paracomplex structure on a conformally flat manifold is parallel, the scalar curvature of the metric must be zero.
Long spacelike embeddings can be approximated by isometric ones.
problem Approximating long embeddings to isometric embeddings in Lorentzian spaces.
method Proving approximation by constructing C1 isometric embeddings. result Long spacelike embeddings can be C0-approximated by C1 isometric embeddings. We provide conditions under which an isometric immersion of a (warped) product of manifolds into a space form must be a (warped) product of isometric immersions.
Random walks on metric spaces embed quasi-isometrically into the space.
problem Embedding random subgroups of metric spaces quasi-isometrically.
method Analyzing random walks and contracting elements in metric spaces.
result Random subgroups of isometry groups are quasi-isometrically embedded.
Study characterizes quasi-isometric embeddings of maps from cusped surfaces into moduli space.
problem Characterizing quasi-isometric embeddings of maps from cusped surfaces into moduli space.
method Investigates shrinking maps from a cusped hyperbolic surface into the moduli space of closed Riemann surfaces, considering quasi-isometric embeddings with respect to Teichmüller distance and intrinsic distance.
result Characterizations of quasi-isometric embeddings are solely determined by the map's monodromy under mild conditions.
Researchers find a list of non-isometric toric para-Kaehler-Einstein manifolds.
problem Finding a complete list of mutually non-isometric Kaehler-Einstein manifolds immersed in a finite-dimensional Kaehler space form.
method Analytical approach to find mutually non-isometric toric para-Kaehler-Einstein manifolds.
result A list of mutually non-isometric toric para-Kaehler-Einstein manifolds analytically immersed in a finite-dimensional para-Kaehler space form.
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 examples and prove results about existence and rigidity.
Study on immersions with flat normal bundle in curved spaces.
problem Behavior of isometric immersions with negative curvature.
method Investigation of second fundamental form growth in space forms.
result Second fundamental form grows exponentially if normal bundle is flat.
Study on isometric submanifolds with preserved Gauss map metrics.
problem Investigating isometric immersions with preserved Gauss map metrics.
method Local analysis of isometric immersions and deformations in Euclidean space.
result Local characterization of non-minimal and non-reducible isometric immersions.
Holomorphic curves in moduli spaces are quasi-isometrically immersed.
problem Understanding the geometric properties of holomorphic curves in moduli spaces.
method Analyzing the quasi-isometric immersion of holomorphic maps from hyperbolic surfaces to moduli spaces.
result Holomorphic curves are quasi-isometrically immersed with parameters depending on surface and moduli space properties.
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.
We prove an analogue of Farb-Masur's theorem that the length-spectra metric on moduli space is "almost isometric" to a simple model V(S) which is induced by the cone metric over the complex of curves. As an application, we know that the Teichmüller metric and the length-spectra metric are "almost isometric…
This paper studies isometric immersions of space forms by means of a hierarchy of finite dimensional integrable systems in Lax form on loop algebras.
Proves local isometric embedding of low-differentiability metrics in 3D space.
problem Isometric embedding of metrics of low differentiability in Euclidean 3-space.
method Simplified notation, geodesic and level parameters, solutions of initial value problems for first order non-linear PDEs, classical linear algebraic systems.
result Local isometric embedding exists for metrics of C1 differentiability.
We study isometric embeddings of C2 Riemannian manifolds in the Euclidean space and we establish that the Hölder space C1,21 is critical in a suitable sense: in particular we prove that for α>21 the Levi-Civita connection of any isometric immersion is induced by the Euclidean connection, wh…
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…
CAT(0) spaces quasi-isometric to Euclidean spaces are homeomorphic to R^n.
problem Characterizing CAT(0) spaces quasi-isometric to Euclidean spaces.
method Analyzing properties of CAT(0) spaces and using quasi-isometry.
result CAT(0) spaces quasi-isometric to R^n are not necessarily homeomorphic to it.
We investigate the structure of real hypersurfaces with isometric Reeb flow in Kaehler manifolds. As an application we classify real hypersurfaces with isometric Reeb flow in irreducible Hermitian symmetric spaces of compact type.
We show that any metric on S2 with Gauss curvature K≥−κ admits a C1,1-isometric embedding into the hyperbolic space with sectional curvature −κ. We also give a sufficient condition for a metric on S2 to be isometrically embedded into anti-de Sitter spacetime with the prescribed cosmological time fun…
Extends submanifold theorem to general spaces.
problem Tackles submanifolds in general ambient spaces.
method Uses development of curves in positive codimension and generalizes Cartan-Ambrose-Hicks theorem.
result Provides a geometric construction of isometric immersions.
Spaces with similar long paths have similar shapes.
problem Comparing shapes of Gromov hyperbolic spaces.
method Examining asymptotic marked length spectra.
result Spaces with identical spectra are roughly isometric.
Hilbert-Efimov theorem states that any complete surface with curvature bounded above by a negative constant can not be isometrically imbedded in R3. We demonstrate that any simply-connected smooth complete surface with curvature bounded above by a negative constant admits a smooth isometric embedding into t…
Estimates for spacelike hypersurfaces in de Sitter space.
problem Isometric embedding problem in de Sitter space.
method Obtained Weyl-type estimates and closedness in the space of embeddings.
result Weyl-type estimates and closedness in the space of embeddings.
Generalizes embedding complex Grassmannians into quadrics.
problem Holomorphic isometric embeddings of complex Grassmannians into quadrics.
method Generalization of do Carmo-Wallach theory for moduli spaces.
result Moduli spaces of embeddings discussed.
Discussing rigidity in codimension 2, extending rigidity concepts.
problem Local isometric rigidity problem in codimension 2.
method Extending rigidity concepts to include genuine and honest rigidity, studying isometric immersions in semi-Euclidean spaces.
result Necessity of natural singularity in inner product for transitivity.
Conditions for reducing quasi-actions to tree actions and group properties.
problem Conditions for reducing quasi-actions to tree actions.
method Reduction to cobounded isometric actions on trees.
result Groups with quasi-orbits quasi-isometric to trees are virtually free.
We prove the equivalence between a relative bottleneck property and being quasi-isometric to a tree-graded space. As a consequence, we deduce that the quasi-trees of spaces defined axiomatically by Bestvina-Bromberg-Fujiwara are quasi-isometric to tree-graded spaces. Using this we prove that mapping class groups quasi-…
Developed a new concept of isometric surfaces in isotropic space.
problem Missing notion of isometric deformations in isotropic space.
method Using Gauss' Theorema Egregium as a condition, developed a new concept.
result Natural analogues of Euclidean space isometries found in isotropic space.
The study proves curvature bounds for quotient spaces of isometric actions.
problem Proving curvature bounds for quotient spaces of isometric actions.
method Disintegrate absolutely continuous measures and define a functional to prove curvature bounds.
result Necessary and sufficient conditions for Ricci curvature to be bounded below.