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Questions tagged [embeddings]

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83 views

Intrinsic Reach for a Riemannian manifold

The reach of a set $X\subseteq \mathbb{R}^d$ is the supremum of all $r \geq 0$ such that for all $y\in X^c$ with $dist(y,X)<r$ there is a unique $x\in X$ with $dist(y,x)= dist(y,X)$. My question: ...
-3
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1answer
188 views

Can there be elementary embedding between a universe and a universe inside it?

[EDIT] the prior question was trivially false, however the intention is to arrange a possible world of such universes, in other words the question is about if it is possible to have a proper class $\...
9
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1answer
394 views

A wild embedding of $\mathbb{S}^1$ into $\mathbb{R}^3$

Can one construct an embedding of $\mathbb{S}^1$ into $\mathbb{R}^3$ so that every orthogonal projection onto a two dimensional plane is a unit disc? It is easy to construct an embedding of $\...
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0answers
215 views

Embedding of $CP^2/CP^1$ into euclidean space [closed]

Is there a "nice" embedding of $\mathbb{C}\mathbb{P}^2/\,\mathbb{C}\mathbb{P}^1$ into $\mathbb{R}^8$?
6
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2answers
125 views

Embedding open connected Riemann Surfaces in $\mathbb{C}^2$

This question arises in the context of a question asked on MSE: Are concrete Riemann surfaces Riemann domains over $\mathbb{C}$. Part of the answer to that question is the question above which is ...
0
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1answer
105 views

$f_n$ is bounded in $C(0,T;H^2(0,L))$ so is $f_n^p$?

Let $1<p<\infty$, and $f_n$ be a bounded sequence in $C(0,T;H^2(0,L))$. It looks obvious to me that $f_n^p$ is also bounded in $C(0,T;H^2(0,L))$. When we take the derivative of $f^p(t)$ twice we ...
12
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1answer
368 views

Embedding Riemannian manifolds into some infinite dimensional manifolds?

First of all I am new to the field of embedding one manifold into another other. I have recently come across with the paper "Embedding Riemannian manifolds by their heat kernel" by P. BERARD, G. ...
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40 views

Theoretical justification of time-series forecasting using Takens' embedding

This is a cross-posting where I couldn't get an answer. In the meantime I have tried to improve the original logic: As in Takens original paper about his embedding theorem, consider a compact $m$-...
1
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1answer
127 views

Trouble with plane embedding

Let $C$ be the middle-thirds Cantor set. Obviously $C\times [0,1]$ embeds into the plane. But $C\times D$ does not, $D$ being a closed disc in the plane. Are there any general results which can be ...
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0answers
84 views

Max-min genus of a bipartite graph

As usual, the genus of a graph with a prescribed circular ordering of the edges at each vertex is defined as the minimum genus of an orientable surface in which the graph can be drawn without edge ...
3
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1answer
225 views

Generalizations of Abhyankar-Moh theorem (embeddings of the line in the plane)

Abhyankar-Moh theorem says that if $L$ is a complex line in the complex affine plane $\mathbb{C}^2$, then every embedding of $L$ into $\mathbb{C}^2$ extends to an automorphism of the plane. It seems ...
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2answers
723 views

When does homology represent an embedded sphere?

If we have a triangulation of a manifold $M$ of dimension $i$ and we have simplicial homology $H_i(M)=\mathbb{Z}$, what is the condition than there exists an embedded sphere $S^i$ that generates the ...
4
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1answer
370 views

Is the space of tangents actually the tangent space?

This is a crosspost of this MSE question. Given a locally Euclidean (locally homeomorphic to some Euclidean space) topological subspace $X\subset\mathbb R^n$ and $p\in X$, let $\mathrm{T}_pX$ denote ...
11
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1answer
207 views

Can $\Delta_1$ (or $\Delta_0$)-elementary embeddings from $V$ to $V$ exist?

Suppose V is a model of Godel-Berney's set theory with the axiom of choice. A well-known result of Kunen says that there can be no elementary embedding $V$ to itself. This result further implies that ...
2
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1answer
229 views

Second fundamental form and embeddings

Let $\Sigma$ be a smooth hypersurface of a $d$ dimensional smooth Riemannian manifold $(\mathcal M, G)$; we may see $G_x$ as a mapping from $T_x(\mathcal M)$ into $T_x^*(\mathcal M)$ so that $$ \...
1
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1answer
96 views

On graph imbedding genus clarification

Given a graph the minimum genus $g$ is the minimum number of handles needed so that there an imbedding of the graph on the surface with no edge crossings. If the graph is of genus $g$ then is there ...
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2answers
140 views

Number of non-equivalent graph embeddings

Given a graph $G$, there is a minimal integer $g$ associated with it which captures the minimum genus a surface needs to have so that $G$ embeds in the surface without edge crossings. Is there a way ...
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0answers
77 views

Is there a quaternionic analogue of Kodaira's embedding theorem?

Let $M$ be a $4m$-dimensional Quaternion-Kähler manifold of positive scalar curvature. Does there exist an $n$ large enough, so that $M$ can be embedded inside $\mathbb{H}P^n$ via a quaternionic ...
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0answers
146 views

Whitney-like embedding theorem for posets?

The Whitney embedding theorem says that any finite-dimensional smooth manifold can be embedded into $\mathbb{R}^n$ for some $n$. Is anything like this true for posets? I'm looking for conditions on a ...
4
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1answer
135 views

Embedding spaces and surface knots in high dimensional manifolds

This is a variation of Craig's Knot complement diffeomorphism groups and embedding spaces for a different type of very simple manifold (surfaces which have a 1-relator fundamental group instead of ...
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0answers
32 views

Realizing set systems in real space

Let $\mathcal{S} = \{S_i\}$ be a collection of subsets of the same size $s$, all drawn from the universe $[n]$, with the property that $|S_i \cap S_j| \le 1$ for all $ i \ne j$. Let us say that $\...
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0answers
294 views

Noncompactness of the Sobolev embedding in the critical exponent case

Let $\Omega \subset \mathbb R^n$ be a bounded domain with a Lipschitz boundary and $n > p \ge 1$. It is well known that up to the critical exponent $p^* = pn/(n − p)$, i.e. $q < p^*$, the ...
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0answers
132 views

Amalagamation of a sequence of closed immersions of schemes

Let $(X_n)_{n \geq 0}$ be a family of schemes. Let $$X_0 \to X_1 \to X_2 \to \dotsc$$ be a sequence of closed immersions (which therefore gives rise to an ind-scheme). Under which (necessarly and/or ...
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0answers
123 views

Intersection of weighted Sobolev spaces

Consider the Sobolev spaces with $p=2$, defined for $s \in \mathbb{R}$ as \begin{equation} W^{s} = \left\{ u \in \mathcal{S}', \ (1 + \lvert \cdot \rvert^2)^{{s}/{2}} \widehat{u} \in L_2 \right\}. \...
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0answers
402 views

Can some exotic sphere be diffeomorphically embedded into some $R^n$?

Can some (or perhaps every) exotic sphere be diffeomorphically embedded into some $R^n$? How does such an embedding (if it exists) look like? I.e., what are the equations for a particular embedding? ...
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2answers
2k views

Questions on J. F. Nash's answer about his errors in the proof of embedding theorem

In the interview of John Nash taken by Christian Skau and Martin Gaussen, in EMS Newsletter, September, 2015 when asked Is it true, as rumours have it, that you started to work on the embedding ...
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2answers
131 views

Normal variation of embedded surfaces [closed]

Let $M^3$ be a complete riemannian manifold and $\Sigma ^2\subset M^3$ a embedded minimal compact surface. Consider the normal variation $\phi: \Sigma \times \Bbb{R}\to M$ given by $$\phi(p,t)=\exp_p(...
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1answer
124 views

Do Morse functions induce embeddings?

Does the existence of Morse functions on smooth manifolds imply Whitney's embedding theorem? (That is, given a smooth manifold $M$, does the existence of a Morse function $f:M \to \mathbb{R}$ imply ...
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0answers
99 views

“Nice” limits of sequences of smooth embeddings

Consider smooth embeddings of a manifold $M$ into some $\mathbb{R}^n$. If a sequence $f_k : M \to \mathbb{R}^n$ of such embeddings converges to some continuous function $f : M \to \mathbb{R}^n$, then ...
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64 views

Obstructions to symplectically embedding compact manifolds of dimension $4$ or higher

It is known in Li's paper (http://arxiv.org/pdf/0812.4929v1.pdf) that in compact symplectic manifolds $(X^{2n},\omega)$ of dimension at least $2n\geq 4$, an immersed symplectic surface represents a $2$...
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1answer
160 views

Isotopy class of closed 2-ball embedded in R^3

My intuition tells me that any two topological embeddings of the closed 2-ball (aka unit disk) into $R^3$ are isotopic in $R^3$. Is this correct? Maybe easy to prove? It seems like it should be easy ...
1
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1answer
101 views

Nash-type theorems for Poisson manifolds

My question comes as a natural follow-up of the previous one which concerned symplectic manifolds: if $(M, P)$ is a Poisson manifold, what embedding theorems are there into some target space (I am ...
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1answer
538 views

Is there a Nash-type theorem for symplectic manifolds?

If $(M, \omega)$ is a symplectic manifold, is it possible to embed (or injectively immerse) it symplectically into a sufficiently large $(\Bbb R ^{2N}, \Omega)$, (with the usual symplectic structure)? ...
9
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1answer
325 views

embedding of quaternionic projective spaces

Let $\mathbb{H}P^m$ be the $m$-th quaternionic projective space. What is the smallest integer $N$ such that there exists an embedding $$ \mathbb{H}P^2\longrightarrow \mathbb{R}^N? $$ Are there any ...
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0answers
90 views

Besov regularity of càdlàg functions?

Let $D(\mathbb{R})$ be the space of functions from $\mathbb{R}$ to $\mathbb{R}$ that are right continuous with left limits (also referred to as càdlàg functions). $D(\mathbb{R})$ is often called the ...
2
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1answer
634 views

embeddings of product of spheres in Euclidean spaces [closed]

I notice that $T^2=S^1\times S^1$ can be embedded in $\mathbb{R}^3$ as a hypersurface (submnaifolds of codimension 1). In general, (1). could the product of spheres $S^{m_1}\times\cdots\times S^{...
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0answers
142 views

obstructions to embeddings of manifolds into Grassmannians

Let $G_k(\mathbb{R}^n)$ be the Grassmannian consisting of $k$-dimensional subspaces in $\mathbb{R}^n$ and $AG_k(\mathbb{R}^n)$ the "affine Grassmannian" consisting of $k$-dimensional planes in $\...
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0answers
180 views

Compact embedding of ${\rm L}^1_{loc}$ space

I was reading one preprint and stumbled upon a part in the proof where one particular embedding was used. Namely: Let $\Omega\subset{\bf R}^2$ be open and bounded and let $p\in\langle 1,2\rangle$. ...
2
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1answer
115 views

VLSI circuit embeddings

In the following paper by Valiant http://www.computer.org/csdl/trans/tc/1981/02/06312176.pdf He shows under theorem 2 (at the bottom of the second page) that any planar graph $G$ of degree 3 or 4 ...
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0answers
188 views

Embeddings between weighted Besov spaces

Consider the Besov spaces $B_{p,q}^s(\mathbb{R}^d)$ for parameters $0<p,q\leq \infty$ and $s\in \mathbb{R}$. The weighted Besov space $B_{p,q}^s(\mathbb{R}^d;\mu)$ is defined for $\mu \in \mathbb{R}...
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0answers
144 views

Equivariant isometric embedding of manifolds in a Hilbert space under a noncompact group action

Given a Riemannian manifold $M$ and a group of isometries $G$ of $M$, I am interested in when there exists a isometric embedding $\iota : M \to H$, where $H$ is a Hilbert space and a representation $\...
0
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1answer
191 views

When is the identity Hilbert-Schmidt between weighted Sobolev spaces?

Set $w(x) = (1 + |x|^2)^{1/2}$ with $|\cdot|$ the Euclidian norm on $\mathbb{R}^n$. For $s,\mu \in \mathbb{R}$, we define the Sobolev space $$H_2^{s}(\mathbb{R}^n) = \left\{f : \lVert f \rVert_{s} := \...
3
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1answer
379 views

Embedding graphs into hyperbolic spaces

Do we know of a characterization as to when does a graph have a "good" embedding into a hyperbolic space? (And does having such an embedding have a spectral or wavelet analysis signature?) I don't ...
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2answers
192 views

Transformations that leave the Plucker embedding of G(2,4) invariant

I am interested in a group of transformations that leave the Plucker embedding of complex Grassmannian $G(2,4)$ into $CP^5$ given by $\lambda_{12}\lambda_{34}-\lambda_{13}\lambda_{24}+\lambda_{14}\...
9
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1answer
215 views

Complements of unknotted tori (higher dimensions)

It is well-known that an unknotted 2-torus in $S^3$ provides the standard Heegaard splitting, in particular its complement consists of two solid tori. It is also known that an unknotted 3-torus in $S^...
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1answer
425 views

Condition to obtain a not compact embedding

I have the two spaces $W_0^{1,p}$ with the norme $$||u||^p=||u||^p_{L^p}+||\nabla u||^p_{L^p}$$ and $$L^{p^*}_{\alpha}=\{ u~\text{measurable}, \int_{\Omega} (|x|^{\alpha} u(x)|)^{p^*} dx<\infty\}$$ ...
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1answer
188 views

Global geometry measures for Riemannian manifolds

I'm working on a stochastic algorithm and considering it to apply in case of any curved space (manifolds). But in order to make the algorithm as efficient as possible I want to include in it some ...
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3answers
541 views

Mathematical value of constructing sphere eversions

I am extremely impressed by the work that has been done constructing sphere eversions, and other similar explicit geometrical proofs. In particular, surely nobody can fail to be impressed by the ...
6
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1answer
288 views

Embedding Euclidean buildings into products of trees

A Euclidean building has a natural metric space structure. (A definition of Euclidean building can be found on Wikipedia, or, more expansively, in Section 4 of Kleiner-Leeb.) Question: Is it true ...
10
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1answer
308 views

Are there non-trivial graphs that uniquely embed to hypercubes?

The $n$-dimensional hypercube is the graph formed by $0$-$1$ sequences of length $n$ where two vertices are adjacent if they differ at only one place. The weight of a sequence is the number of $1$'s ...