Nikolai Nadirashvili, Yu Yuan
We construct smooth metrics on 2-manifold with nonpositive Gauss curvature which cannot be (C^3) locally isometrically embedded in R^3. Moreover, the Gauss curvature of the metric can be made negative except for one point.
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Counterexamples to the Lorentzian Calderón problem
研究显示非等距的光滑全球超曲面洛伦兹度量在无限圆柱体上具有相同的超几何狄利克雷到神经元映射
An complete exposition of Matthias Gunther's elementary proof of Nash's isometric embedding theorem.
In this work we prove the fact that, for a short time, it is possible to construct a smooth parametrized family of isometric embeddings of an arbitrary smooth parametrized family of Riemannian metrics on a smooth closed manifold into an Euclidean space. In order to prove this statement we work out stability estimates within Matthias Günther's local perturbation method to derive a time-dependent local perturbation method around a free isometric embedding. Iteratively, we use this time-dependent local perturbation method to construct the desired family of isometric embeddings.
We study the problem of isometrically embedding a two-dimensional Riemannian manifold into Euclidean three-space. It is shown that if Gaussian curvature vanishes to finite order and its zero set consists of two smooth curves tangent at a point, then local sufficiently smooth isometric embedding exists.
We study isometric embeddings of $C^2$ Riemannian manifolds in the Euclidean space and we establish that the Hölder space $C^{1,\frac{1}{2}}$ is critical in a suitable sense: in particular we prove that for $α> \frac{1}{2}$ the Levi-Civita connection of any isometric immersion is induced by the Euclidean connection, whereas for any $α< \frac{1}{2}$ we construct $C^{1,α}$ isometric embeddings of portions of the standard $2$-dimensional sphere for which such property fails.
We study the old problem of isometrically embedding a 2-dimensional Riemannian manifold into Euclidean 3-space. It is shown that if the Gaussian curvature vanishes to finite order and its zero set consists of two Lipschitz curves intersecting transversely at a point, then local sufficiently smooth isometric embeddings exist.
We construct counterexamples to inverse problems for the wave operator on domains in $\mathbb{R}^{n+1}$, $n \ge 2$, and on Lorentzian manifolds. We show that non-isometric Lorentzian metrics can lead to same partial data measurements, which are formulated in terms certain restrictions of the Dirichlet-to-Neumann map. The Lorentzian metrics giving counterexamples are time-dependent, but they are smooth and non-degenerate. On $\mathbb{R}^{n+1}$ the metrics are conformal to the Minkowski metric.
Hilbert-Efimov theorem states that any complete surface with curvature bounded above by a negative constant can not be isometrically imbedded in $\mathbb{R}^3.$ We demonstrate that any simply-connected smooth complete surface with curvature bounded above by a negative constant admits a smooth isometric embedding into the Lorentz-Minkowski space $\mathbb{R}^{2,1}$.
In this paper we prove a conjecture of Bryant, Griffiths, and Yang concerning the characteristic variety for the determined isometric embedding system. In particular, we show that the characteristic variety is not smooth for any dimension greater than 4. This is accomplished by introducing a smaller yet equivalent linearized system, in an appropriate way, which facilitates analysis of the characteristic variety.
On the isometric version of Whitney's strong embedding theorem
本文在Nash-Kuiper h-原理框架下，证明了任意n维光滑紧致流形可等距嵌入2n维欧氏空间，且嵌入具有特定Hölder正则性，方法为凸积分与粘合技术。
We establish the weak continuity of the Gauss-Coddazi-Ricci system for isometric embedding with respect to the uniform $L^p$-bounded solution sequence for $p>2$, which implies that the weak limit of the isometric embeddings of the manifold is still an isometric embedding. More generally, we establish a compensated compactness framework for the Gauss-Codazzi-Ricci system in differential geometry. That is, given any sequence of approximate solutions to this system which is uniformly bounded in $L^2$ and has reasonable bounds on the errors made in the approximation (the errors are confined in a c…
$C^{1,1-ε}$ Isometric embeddings
本文通过凸积分与额外迭代方法，在余维数为 $\ rac{1}{2}n(n+1)$ 的情形下建立了 $n$ 维流形 $C^{1,1-ε}$ 等距嵌入的局部 $h$-原理。
We give a new proof for the local existence of a smooth isometric embedding of a smooth $3$-dimensional Riemannian manifold with nonzero Riemannian curvature tensor into $6$-dimensional Euclidean space. Our proof avoids the sophisticated arguments via microlocal analysis used in earlier proofs. In Part 1, we introduce a new type of system of partial differential equations, which is not one of the standard types (elliptic, hyperbolic, parabolic) but satisfies a property called strong symmetric positivity. Such a PDE system is a generalization of and has properties similar to a system of ordinar…
In this paper we study the embedding of Riemannian manifolds in low codimension. The well-known result of Nash and Kuiper says that any short embedding in codimension one can be uniformly approximated by $C^1$ isometric embeddings. This statement clearly cannot be true for $C^2$ embeddings in general, due to the classical rigidity in the Weyl problem. In fact Borisov extended the latter to embeddings of class $C^{1,α}$ with $α>2/3$. On the other hand he announced in that the Nash-Kuiper statement can be extended to local $C^{1,α}$ embeddings with $α<(1+n+n^2)^{-1}$, where $n$ is the dimension …
We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampere type. These two problems are: the local isometric embedding problem for two-dimensional Riemannian manifolds, and the problem of locally prescribed Gaussian curvature for surfaces in R^3. We prove a general local existence result for a large class of Monge-Ampere equations in the plane, and obtain as corollaries the existence of regular solutions to both problems, in the case that the Gaussian curvature vanishes to arbitrary finite order on a single smooth cu…
In this paper, we are concerned with the existence of local isometric embeddings into Euclidean space for analytic Riemannian metrics $g$, defined on a domain $U\subset \mathbf{R}^n$, which are singular in the sense that the determinant of the metric tensor is allowed to vanish at an isolated point (say the origin). Specifically, we show that, under suitable technical assumptions, there exists a local analytic isometric embedding $u$ from $(U',Π^*g)$ into Euclidean space $\mathbf{E}^{(n^2+3n-4)/2}$, where $Π:U' \to U\backslash\{0\}$ is a finite Riemannian branched cover of a deleted neighborho…
This short note is a mostly expository article examining negatively curved three-manifolds. We look at some rigidity properties related to isometric embeddings into Minkowski space. We also review the Cross Curvature Flow (XCF) as a tool to study the space of negatively curved metrics on hyperbolic three-manifolds, the largest and least understood class of model geometries in Thurston's Geometrisation. The relationship between integrability and embedability yields interesting insights, and we show that solutions with fixed Einstein volume are precisely the integrable solutions, answering a que…
We prove a priori bounds for the trace of the second fundamental form of a $C^4$ isometric embedding into $R^{n+1}$ of a metric $g$ of non-negative sectional curvature on $S^n$, in terms of the scalar curvature, and the diameter of $g$. These estimates give a bound on the extrinsic geometry in terms of intrinsic quantities. They generalize estimates originally obtained by Weyl for the case $n=2$ and positive curvature, and then by P. Guan and the first author for non-negative curvature and $n=2$. Using $C^{2,α}$ interior estimates of Evans and Krylov for concave fully nonlinear elliptic partia…
Given a smooth 2-dimensional Riemannian or pseudo-Riemannian manifold $(M, \boldsymbol{g})$ and an ambient 3-dimensional Riemannian or pseudo-Riemannian manifold $(N, \boldsymbol{h})$, one can ask under what circumstances does the exterior differential system $\mathcal{I}$ for the isometric embedding $M\hookrightarrow N$ have particularly nice solvability properties. In this paper we give a classification of all $2$-metrics $\boldsymbol{g}$ whose local isometric embedding system into flat Riemannian or pseudo-Riemannian 3-manifolds $(N, \boldsymbol{h})$ is Darboux integrable. As an illustratio…
The present article studies holomorphic isometric embeddings of the complex two--plane Grassmannnian into quadrics. We discuss the moduli space of these embeddings up to gauge and image equivalence using a generalisation of do Carmo--Wallach theory.
A dynamical--topological obstruction for smooth isometric embeddings of Riemannian manifolds via incompressible Euler equations
本文提出带边 Riemannian 流形等距嵌入的一个动力学--拓扑障碍，证明在第一实同调非平凡、基本群中心平凡且嵌入维数至少为 3 时，等距嵌入必然违背刚性同痕延拓性质，论证受不可压缩 Euler 方程启发。
This note is about a little extension of Nash's embedding theorem in the case of complete manifolds.
A proof of the isometric embedding of a given two-metric in E^3 of class C^1. The method uses the theory of first order partial differential equations. The curvature of the metric plays no role in the proof.
The present article studies holomorphic isometric embeddings of arbitrary complex Grassmannians into quadrics, generalising results in [13]. The moduli spaces of these embeddings up to gauge and image equivalence are discussed using a generalisation of do Carmo-Wallach theory.
In this paper, we construct smooth isometric embeddings of multiple warped product manifolds in quadrics of semi-Euclidean spaces. Our main theorem generalizes previous results as given by Blanusa, Rozendorn, Henke and Azov.
A counterexample to Yau's bounded-mean-curvature embedding question
针对Yau关于有界Ricci曲率与正单射半径流形是否存在有界平均曲率等距嵌入的问题，构造了四维及以上维度的反例，证明此类嵌入不存在。
In this article, we study holomorphic isometric embeddings between bounded symmetric domains. In particular, we show the total geodesy of any holomorphic isometric embedding between reducible bounded symmetric domains with the same rank.
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.
In the special case of S^1 invariant metrics on S^2, we find necessary and sufficient conditions for the existence of isometric embeddings into the canonical R^3, in other words: a Weyl type theorem with converse.
We show that any metric on $S^2$ with Gauss curvature $K \geq -κ$ admits a $C^{1,1}$-isometric embedding into the hyperbolic space with sectional curvature $-κ$. We also give a sufficient condition for a metric on $S^2$ to be isometrically embedded into anti-de Sitter spacetime with the prescribed cosmological time function.
We discuss holomorphic isometric embeddings of the projective line into quadrics using a generalisation of the theorem of do Carmo--Wallach to provide a description of their moduli spaces up to image and gauge--equivalence. Moreover, we show rigidity of the real standard map from the projective line into quadrics.
J. Nash proved that the geometry of any Riemannian manifold M imposes no restrictions to be embedded isometrically into a (fixed) ball B_{\mathbb{R}^{N}}(1) of the Euclidean space R^N. However, the geometry of M appears, to some extent, imposing restrictions on the mean curvature vector of the embedding.
We prove that the isometric embedding of any metric of differentiability class C1 in E3 exists. We use simplified notation for the given metric, namely geodesic parameters, and level parameters for the embedded surface in E3. Central to our discussion will be solutions of initial value problems for two first order non-linear partial differential equations. We also make use of the classical theory of linear algebraic systems. We will prove local isometric embedding. An example is given for which the Gaussian curvature of the metric is equal to one but the embedded surface is non-analytic.
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.
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 thus extend earlier rigidity results about quasi-isometric embeddings to the setting of semisimple Lie groups. We also present some examples when the rigidity does not hold, including first examples in which every flat is mapped into multiple flats.
We construct isometric and conformally isometric embeddings of some gravitational instantons in $\mathbb{R}^8$ and $\mathbb{R}^7$. In particular we show that the embedding class of the Einstein--Maxwell instanton due to Burns is equal to $3$. For $\mathbb{CP}^2$, Eguchi--Hanson and anti-self-dual Taub-NUT we obtain upper and lower bounds on the embedding class.
We review classical approaches to the problem of isometrically embedding a Riemannian surface into Euclidean 3-space, including coordinate-based approaches exposited by Darboux and Eisenhart, as well as the moving-frames based approaches advocated by Cartan. In particular, the first approach involves reducing the problem to solving a single PDE; settling the question of when this PDE is integrable by the method of Darboux is the subject of this short note. This is surprisingly easy, since it is related to the analogous question for the isometric embedding system arising from the moving frames …
This article uses Cartan-Kähler theory to construct local conservation laws from covariantly closed vector valued differential forms, objects that can be given, for example, by harmonic maps between two Riemannian manifolds. We apply the article's main result to construct conservation laws for covariant divergence free energy-momentum tensors. We also generalize the local isometric embedding of surfaces in the analytic case by applying the main result to vector bundles of rank two over any surface.
This text is the extended version of a talk given at the conference Geometry, Topology, QFT and Cosmology hold from May 28 to May 30, 2008 at the Observatoire de Paris. Using exterior differential systems, I provide a positive answer to the generalized isometric embedding problem of vector bundles, and show how conservation laws for a class of PDE can be constructed, for instance, for covariant divergence-free energy-momentum tensors.
$C^{1}$-isometric embeddings of Riemannian spaces in Lorentzian spaces
本文证明任意紧致黎曼流形到洛伦兹流形的长空型嵌入可被$C^{1}$等距嵌入在$C^{0}$意义下逼近。
In this article, we study a free boundary isometric embedding problem for abstract Riemannian two-manifolds with the topology of the disc. Under the assumption of positive Gauss curvature and geodesic curvature of the boundary being equal to one, we show that any such disc may be isometrically embedded into the Euclidean three space $\mathbb{R}^3$ such that the image of the boundary meets the unit sphere $\mathbb{S}^2$ orthogonally. Moreover, we also show that the embedding is unique up to rotations and reflections through planes containing the origin. Finally, we define a new Brown-York type …
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 isometrically embedded, even in the case where (M,g) is homogeneous. In this paper, when the space (M,g) is locally isometric to a three-dimensional Lie group equipped with a left-invariant Riemannian metric, we classify all such spaces that can be locally isometrically embedded into…
This text is the extended version of a talk given at 6th Meeting of Integrable Systems and Quantum Filed Theory at Peyresq hold from June 10 2006 to June 17, 2006 at Peyresq, France. The goal of this lecture is to give a brief introduction to Cartan-Kähler's theory. As examples to the application of this theory, we choose the local isometric and conformal embedding. We provide lots of details and explanations of the calculation and the tools used.
On the geometry of Riemannian isometric embeddings
针对与$\RR^n$微分同胚且允许Bieberbach群$Γ$等距作用的黎曼流形$M$，研究其到欧几里得空间有界子集及$Γ$-等变等距嵌入问题，通过新用已知技巧得到嵌入维数结果，并证明n维光滑流形到$\RR^{2N}$有界子集的嵌入性。
On equivariant isometric embeddings of Riemannian manifolds with symmetries
本文研究了具有对称性的黎曼流形的等变等距嵌入问题，证明了存在一个到欧几里得空间的嵌入，其维度与Matthias Günther的结果一致。
In the first part of this paper, we extend the result of Li-Wang on the linearized embedding problem to a compact manifold of arbitrary dimension. Using this, we then show that any metric perturbation of a embedded $n$-sphere is also isometrically embedded up to a solution of the homogenous Codazzi equation, irrespective of the ambient geometry. In the second part we specialize to dimension two, and study these results within an ambient Null Cone. Specifically, given a path of metrics on the 2-sphere and an initial isometric embedding, we develop a small parameter existence and uniqueness theo…
We define inductively isometric embeddings of $\mb{P}^n(\mb{R})$ and $\mb{P}^n(\mb{C})$ (with their canonical metrics conveniently scaled) into the standard unit sphere, which present the former as the restriction of the latter to the set of real points. Our argument parallels the telescopic construction of $\mb{P}^\infty(\mb{R})$, $\mb{P}^\infty(\mb{C})$, and $\mb{S}^\infty$ in that, for each $n$, it extends the previous embedding to the attaching cell, which after a suitable renormalization makes it possible for the result to have image in the unit sphere.
We explore the practicability of Nash's Embedding Theorem in vision and imaging sciences. In particular, we investigate the relevance of a result of Burago and Zalgaller regarding the existence of isometric embeddings of polyhedral surfaces in $\mathbb{R}^3$ and we show that their proof does not extended directly to higher dimensions.
On the isometric embedding problem for length metric spaces
本文证明每个真n维长度度量空间可近似等距嵌入洛伦兹空间$\mathbb{R}^{3n+6,1}$，即在预定测地线集上保持能量泛函。
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. A key ingredient in our proofs of rigidity results is a direct generalization of the Mostow-Morse Lemma in higher rank. Typically this lemma is replaced by the quasi-flat theorem which says that ma…
Isometric Embeddings of Conformally Compact Manifolds into Hyperbolic Spaces
本文证明了共形紧致流形可以等距嵌入到高维双曲空间中，解决了在无穷远处具有负曲率下界流形的嵌入问题。
Let $G$ and $G'$ be simple Lie groups of equal real rank and real rank at least $2$. Let $Γ<G$ and $Λ< G'$ be non-uniform lattices. We prove a theorem that often implies that any quasi-isometric embedding of $Γ$ into $Λ$ is at bounded distance from a homomorphism. For example, any quasi-isometric embedding of $SL(n,\mathbb Z)$ into $SL(n, \mathbb Z[i])$ is at bounded distance from a homomorphism. We also include a discussion of some cases when this result is not true for what turn out to be purely algebraic reasons.
The mathematical theory of isometric embedding is applied to study the notion of quasilocal mass in general relativity. In particular, I shall report some recent progress of quasilocal mass with reference to a cosmological spacetime, such as the de Sitter or the Anti-de Sitter spacetime, or a blackhole spacetime, such as the Schwarzschild spacetime. This article is based on joint work with Po-Ning Chen, Ye-Kai Wang, and Shing-Tung Yau.
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.
Let M be a smooth compact connected oriented manifold of dimension at least two endowed with a volume form. Assuming certain conditions on the fundamental group $π_1(M)$ we construct quasi-isometric embeddings of either free Abelian or direct products of non-Abelian free groups into the group of volume preserving diffeomorphisms of M equipped with the L^p metric induced by a Riemannian metric on M.
We prove that certain Riemannian manifolds can be isometrically embedded inside Calabi-Yau manifolds. For example we prove that given any real-analytic one parameter family of Riemannian metrics $g_t$ on a 3-dimensional manifold $Y$ with volume form independent of $t$ and with a real-analytic family of nowhere vanishing harmonic one forms $θ_t$, then $(Y, g_t)$ can be realized as a family of special Lagrangian submanifolds of a Calabi-Yau manifold $X$. We also prove that certain principal torus bundles can be equivariantly and isometrically embedded inside Calabi-Yau manifolds with torus actio…
We find necessary and sufficient conditions for existence of a locally isometric embedding of a vacuum space-time into a conformally-flat 5-space. We explicitly construct such embeddings for any spherically symmetric Lorentzian metric in $3+1$ dimensions as a hypersurface in $R^{4, 1}$. For the Schwarzschild metric the embedding is global, and extends through the horizon all the way to the $r=0$ singularity. We discuss the asymptotic properties of the embedding in the context of Penrose's theorem on Schwarzschild causality. We finally show that the Hawking temperature of the Schwarzschild metr…
Convolutions are fundamental elements in deep learning architectures. Here, we present a theoretical framework for combining extrinsic and intrinsic approaches to manifold convolution through isometric embeddings into tori. In this way, we define a convolution operator for a manifold of arbitrary topology and dimension. We also explain geometric and topological conditions that make some local definitions of convolutions which rely on translating filters along geodesic paths on a manifold, computationally intractable. A result of Alan Turing from 1938 underscores the need for such a toric isome…
We prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash $C^1$ Embedding Theorem. For more general metric spaces the same result is false, e.g., for Finsler non-Riemannian manifolds. However, we also show that any metric space of finite packing dimension can be embedded in some Euclidean space via a Lipschitz map.
In this article, we study local holomorphic isometric embeddings from ${\BB}^n$ into ${\BB}^{N_1}\times... \times{\BB}^{N_m}$ with respect to the normalized Bergman metrics up to conformal factors. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic map between complex Euclidean spaces by the algebraic extension theorem derived along the lines of Mok and Mok-Ng. Applying holomorphic continuation and analyzing real analytic subvarieties carefully, we show that each component is either a constant map or a proper holomorphic map…
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