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 ordinary differential equations with a regular singular point. A local existence theorem is then established by using a novel local-to-global-to-local approach. In Part 2, we apply this theorem to prove the local existence result for isometric embeddings.
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 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.
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…
A priori bounds for co-dimension one isometric embeddings
证明了在$S^n$上具有非负截面曲率的度量$g$的等距嵌入到$R^{n+1}$中的$C^4$的第二基本形式的迹的先验界，即关于$g$的标量曲率和直径的先验界.这些估计给出了一个约束的外在几何的内在量。他们推广了最初由Weyl获得的$n=2$和正曲率的估计，然后由P.Guan和第一作者获得的非负曲率和$n=2$的估计。利用Evans和Krylov对凹的完全非线性椭圆型偏微分方程的$C^{2,α}$内部估计，这些界允许我们得到以下收敛定理：对于任何$ε>0$，在欧氏空间$R^{n+1}$中等距嵌入的以$ε$为界的非负截面曲率和标量曲率的度量集在Hölder空间$C^{4,α}$，$0<α<1$中是封闭的。这些结果是为了理解我们提出的Weyl嵌入问题的以下高维版本而获得的：\emph{Suppose that $g$ is a smooth metric of non-negative sectional curvature and positive scalar curvature on §^n$ which is locally isometrically embeddable in $R^{n+1}$. Does $(S^n,g)$ then admit a smooth global isometric embedding into $R^{n+1}$?}
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 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 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.
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.
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 …
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 …
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.
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.
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…
本文确定了与非紧型对称空间拟等距的生成群的结构，并允许Euclidean de Rham因子.若$X$是不含Euclidean de Rham因子的非紧型对称空间，且$\Ga$是与乘积$\E^k\times X$拟等距的生成群，则存在精确序列$1\ra H\ra\Ga\ra L\ra 1$，其中$H$包含$\Z^k$的有限索引拷贝，并且$L$是$X$的等距群中的均匀格。
Instituto de Matemáticas, Universidad Nacional Autónoma de México（墨西哥国立自治大学数学研究所）
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…
