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Questions tagged [submanifold]

In mathematics, a submanifold of a manifold $M$ is a subset $S$ which itself has the structure of a manifold, and for which the inclusion map $S \rightarrow M$ satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions. Therefore, please include the details when using this tag.

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Question about curves in integral manifold associated with an involutive distribution

Let $K$ be an integral distribution on a smooth manifold $N$ and $\gamma(t)$ a smooth curve on $N$. If $M$ is an integral manifold of $K$, $\gamma(t)\subset M$ then how to show $\gamma'(t)\in T_{\...
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Having defined submanifolds only, can one talk about manifolds?

I was going through this paper about the Laplace-Beltrami-Operator on a Riemannian manifold: https://pdfs.semanticscholar.org/4fda/3faf5237d0d98051e79477095fafb07076e4.pdf On page 2 he introduces ...
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Why is an inclusion map from an open subset smooth?

If $B$ is a manifold and $A\subseteq B$ is a regular submanifold of $B$, then the inclusion map $i:A\to B$ is an embedding and thus smooth. If $B$ is a manifold, and $A\subseteq B$ is an open subset ...
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34 views

When is an inclusion map smooth?

If $B$ is a manifold, and $A\subseteq B$ is a regular submanifold, then the inclusion map $i:A\to B$ is an embedding and thus smooth. That $i$ is an embedding seems a bit strong. Is there another way ...
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1answer
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Why is $(U \cap R, \varphi_R)$ a chart for a regular submanifold $R$?

Let $R$ be a regular submanifold of a manifold $M$ where $\dim(M)=m$ and $\dim(R)=k$. By definition, for every $r \in R$, there is a chart (in $M$) about p, $(U,\varphi) = (U,x^1,...,x^k,...,x^m)$, ...
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53 views

How Klein Bottle in R⁴ does not have self intersection?

This is the screenshot from Feko's book on Differential Geometry. In 1.5.10 in the given hint how he deduced that in R^4 the Klein Bottle has no self intersection? Also,in 1.5.11, what does square ...
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1answer
20 views

Matrices and Manifolds

I have a huge problem solving this question. Does anybody have literature on this or maybe a solution? Let $ M:=\{x\in \mathbb{R}^n : x^{⊤} A x = r\} \subset \mathbb{R}^n $ for $r>0$ and $A$ ...
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23 views

Motivation usage of Gramian Matrix for Integration on Submanifolds

I am struggling to understand the motivation for the definition of the Gram Matrix and its corresponding role for the definition of the Lebesgue measure on submanifolds. $M \subset \mathbb{R}^n\ k$ ...
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2answers
38 views

matrixgroup is a differentiable submanifold of $M_4(\mathbb{R})$

My question is already existtent Prove that $\mathcal{O}(3,1)$ is a submanifold of $\mathcal{M}_4(\mathbb{R})$ , but $I$ don't understand the hint and there is no answer (and the question there is not ...
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1answer
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How to find the set of all points in a submanifold minimizing the distance to a given point?

I have the following set $M$ defined as the set of all points $(x,y,z)$ such that $ x^2+y^2-z^2=-1$. And I'm asked the following three questions : a) Prove that this is a submanifold of $R^3$ b) Find ...
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2answers
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If $f : M \to N$ is $C^k$, then $\Gamma(f)$ is a closed embedded submanifold of $M \times N$

The following problem is part of an assignment in my Differential Topology course. Exercise. Show the graph of a $C^k$ function $f : M \to N$ between two $C^k$ manifolds $M$ and $N$ defined by $\...
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coordinate systems produce submanifolds

In his book "Semiriemannian geometry with applications to relativity", Barrett O'Neil says on page 16 under definition 26 that "coordinate systems produce submanifolds.If T:U->R^n is a coordinate ...
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Anti-de Sitter space: Is the universal cover of $\text{AdS}_n$ a submanifold of $\text{AdS}_{n+1}$?

$$\text{AdS}_n = \{\vec x\in\mathbb R^{n-1,2}\mid\vec x\cdot\vec x=-1\}$$ Using an orthonormal basis $\{e_i\}$ for $\mathbb R^{n-1,2}$, with ${e_1}^2={e_2}^2=-1$ and $(e_{\geq3})^2=1$, this space can ...
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1answer
62 views

If $S\subseteq M$ is a submanifold, is there a canonical way to identify $T_{p}^{*}S$ as a subspace of $T_{p}^{*}M$?

I have a few questions. Any thoughts to any one of them will be appreciated. Suppose $S\subseteq M$ is an embedded submanifold of $M$. There is a convenient characterization of the tangent spaces of $...
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1answer
37 views

$U(n)$ and $SU(n)$ are connected smooth submanifold of $M_{n,n}(\mathbb{C})$

How can I prove that $U_n(\mathbb{C})$ and $SU_n(\mathbb{C})$ are smooth submanifolds of $M_{n,n}(\mathbb{C})$ ? I know that given the manifold $X$, $Y$ is a smooth submanifold of $X$ if $\forall k$ ...
2
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1answer
39 views

Show that $SO(3)$ in an embedded submanifold of $\mathbb{R}^{3 \times 3}$

How does one go about proving that $SO(3)$ is an embedded submanifold of $\mathbb{R}^{3 \times 3}$? I know that there is a definition of embedded submanifold in terms of the induced topology, and I ...
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34 views

Why is the submanifold a Nullset?

I tried to explain myself why a sumbanifold of the dimension $k<n$ is a Nullset in $\mathbb{R}^n$. May this be explained by the fact, that the submanifold is locally coverable by the pictures of ...
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1answer
95 views

Graph is an Embedded Submanifold

I'm trying to work through the following problem. Let $M$ be an $m$-dimensional manifold. Prove that the graph of any smooth map $f:M\to\mathbb{R}$ is a closed, embedded submanifold of $M\times\...
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2answers
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Why $\mathbb R$ with the chard $t\longmapsto t^3$ is not a submanifold of $\mathbb R^2$ ?

I'm not very used to Manifold. I'm a little bit confuse with something. 1) Let $\phi:t\longmapsto t^3$ with $t\in \mathbb R$. So $(\mathbb R,\phi)$ is a smooth manifold. Now, why this is not a ...
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Regular level set and submanifold

I read a textbook and it says: A level set $S = \{x:F(x)=c\}$, $F: \mathbb{R}^n\rightarrow\mathbb{R}^k$ being analytic. $S$ is regular if it is nonempty and the Jacobian matrix of $F$ has maximal ...
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1answer
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Why the 'gradient of the diffeomorphism at a point in the surface' perpendicular to the surface at that point?

This question is related to these two questions of mine: Intuition or motivation for the definition of an hypersurface. What are we actually trying to define? and Understanding this very generic ...
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26 views

Show that a map is an embedding

I guess that the image (all matricies of rank $2r$ with $r$ positive and $r$ negative eigenvalues) of the map $$\psi: GL_n(\mathbb{C}) \to M_{n,n}(\mathbb{C}) \qquad A \mapsto A(-1_r \oplus {1}_r \...
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1answer
96 views

Existence of Global Defining Function for Hypersurface

Let $M$ be a smooth manifold, and $\Sigma$ a hypersurface of $M$. (That is, $\Sigma$ is smoothly a embedded subset of $M$ with codimension $1$.) By a defining function for $\Sigma$, we mean some $f \...
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1answer
78 views

How bad can the intersection of two totally geodesic submanifolds be?

Let $M$ be a complete Riemannian manifold and let $S_1,S_2 \subset M$ be totally geodesic submanifolds which are closed as subspaces of $M$. Question: Is $S_1 \cap S_2$ a submanifold of $M$? Or is it ...
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1answer
49 views

Submanifold of $S^1 \times S^1 \times \mathbb{R}^2$. [closed]

Let's $f,g : S^1 \longrightarrow \mathbb{R}^2$ be embeding's. Consider $F: S^1 \times S^1 \times \mathbb{R}^2 \longrightarrow \mathbb{R}^2 $, $F(x,y,v)= f(x) - g(y) -v$. Show that $N= F^{-1}(0)$ is ...
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1answer
57 views

“Lift” of an immersion

Let $f : \Sigma^k \to M^n$ be an immersion between differentiable manifolds and let $\pi : \tilde{M} \to M$ be a finite-to-one covering map. Let $\tilde{\Sigma} = \pi^{-1}(f(\Sigma))$. Is it true that ...
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2answers
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Tangent Space to the Quadratic Form

Let $S:=\{x \in \mathbb{R}^n : x^\top Ax = 1 \}$. We know that $S$ is a $(n-1)$-dimensional submanifold of $\mathbb{R}^n$ because it is a regular level set of a smooth function. What is the tangent ...
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1answer
53 views

$T^1M = \{(p,v) \in TM; \| v\| = 1 \}$ is a submanifold of $TM$ of dimension $2n-1$

Let $M$ submanifold of $\mathbb{R}^n$. Show that $T^1M = \{(p,v) \in TM; \| v\| = 1 \}$ is a submanifold of $TM$ of dimension $2n-1$. Comments: I'm trying to build the function $$f: TM \...
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1answer
65 views

Differentiability of functions on Manifolds

Let $f:\mathbb{R}^n\to\mathbb{R}^m$ be smooth, and suppose $M$ and $N$ are submanifolds of $\mathbb{R}^n$ and $\mathbb{R}^m$. Moreover, assume that $f(M) \subseteq N$. I want to prove that $f:M\to N$ ...
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1answer
37 views

Existence of a unique “outer” normal vector

A compact set $K\subseteq\mathbb R^3$ is said to have a smooth boundary, if for all $p\in\partial K$ there is an open neighborhood $U$ of $p$ and a continuously differentiable function $\psi:U\to\...
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2answers
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How can we show that a nondegenerate “open” triangle is a $2$-dimensioal $C^\infty$-submanifold of $\mathbb R^3$?

How can we show that a nondegenerate triangle $\Delta$ (without the edges) is a $2$-dimensioal $C^\infty$-submanifold of $\mathbb R^3$? Intuitively, the claim is obvious to me: For any point $p$ of $\...
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1answer
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How is the “surface measure” on a manifold defined?

Let $k,n\in\mathbb N$ with $k\le n$ $M$ be a $k$-dimensional $C^1$-submanifold of $\mathbb R^n$ $\Omega\subseteq\mathbb R^k$ be open, $\phi:\Omega\to M$ be a global chart of $M$ and $$g_\phi(x):=\det\...
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1answer
61 views

Is “immersion by parts” on a union of submanifolds an immersion?

Let $M$ be a smooth manifold of dimension greater than $2$. Suppose $H_1,H_2,\dots,H_k$ are disjoint embedded submanifolds of $M$. Suppose that $H_k$ is open and dense in $M$, and that $\cup_{i=1}^k ...
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1answer
100 views

Is the image of an equivariant map always a weakly embedded submanifold?

Let $M,N$ be smooth manifolds, with a smooth $G$-action on them, by some Lie group $G$. Suppose also that $M$ has a finite number of orbits under $G$'s-action. Let $f:M \to N$ be a smooth, ...
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0answers
8 views

Differentiating functions defined on level sets

For $ M, N $ manifolds, $ s : M, N $ a submersion and $ S = s^{-1}(\{y\}) $ a regular submanifold of $ M $, is there any way to recover the "intrinsic" tangent space of $ S $ and differentiate ...
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1answer
69 views

Is the image of the map $A \to \bigwedge^{k}A $ from matrices above rank $k$ closed?

$\newcommand{\Cof}{\operatorname{cof}}$ $\newcommand{\id}{\operatorname{Id}}$ $\newcommand{\End}{\operatorname{End}}$ $\newcommand{\GL}{\operatorname{GL}}$ Let $V$ be a real $d$-dimensional vector ...
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2answers
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Is the image of the map $A \to \bigwedge^{k}A $ an embedded submanifold of $\text{GL}(\bigwedge^{k}V)$?

$\newcommand{\Cof}{\operatorname{cof}} \newcommand{\id}{\operatorname{Id}}$ Let $V$ be a real oriented $d$-dimensional vector space ($d>2$). Let $2 \le k \le d-1$ be fixed. Consider the following ...
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1answer
117 views

The canonical projection $\pi: \nu M \rightarrow M$ is a submersion

Let $M$ be a subvariety of $\mathbb{R}^n$. Show that the canonical projection $\pi: \nu M \rightarrow M$ is a submersion. Where $\nu M=\{(x,v)\in \mathbb{R}^n \times\mathbb{R}^n;x\in M \text{ and } v\...
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1answer
102 views

What is the relation between totally real submanifold and Lagrangian submanifold?

By definition, for a complex manifold M, totally real submanifold X of M is satisfying 1) $2 dim X$ = $dim M$ and 2) $T_pM \cap J T_pM =\{0\}$ for $\forall p \in X $with integrable complex ...
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2answers
137 views

Tangent bundle $TM$ is a submanifold of $\mathbb{R}^{2n}$

Let $M$ be a submanifod of $\mathbb{R}^n$. Show that the tangent bundle $TM$ is a submanifold of $\mathbb{R}^{2n}$. ps.:I'm trying to use charts, but I'm slowing down, because I've been studying this ...
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2answers
104 views

Map $F$ open and $F(\mathbb{R})$ is a submanifold

Let $F: \mathbb{R} \rightarrow S^1 \times S^1$ defined by $$F(t) = ((\cos2\pi t, \sin2 \pi t), (\cos2\pi \alpha t, \sin2 \pi \alpha t))$$. For which values ​​of $\alpha$ the map $F$ is open? For ...
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1answer
47 views

If $F$ is a $1$-to-$1$ imersion and is proper then $F$ is an imbedding

Let $F:N\rightarrow M$ be a one-to-one immersion which is proper, i.e. the inverse image of any compact set is compact. Show that $F$ is an imbedding and that its image is a closed regular submanifold ...
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1answer
77 views

Inequality about dimensions of submanifolds

Assume $M1,M2\subseteq \mathbb R^n$ are two smooth submanifolds such that $M1\subseteq M2$. How can I prove that $dim(M1)\le dim(M2)$? What I know: $M \subseteq \mathbb R^n$ is a k-dimensional ...
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1answer
72 views

$Sp(2n)$ is embedded in $GL(2n)$ and has dimension $2n^2+n$

Let $Sp(2n):=\{A\in\mathbb{R}^{2n\times 2n}\mid A^tA_0A=A_0\}$ be the group of symplectomorphisms from $(\mathbb{R}^{2n},\omega_0)$ to itself, where: \begin{gather} A_0 := \begin{bmatrix}{} 0 &...
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0answers
22 views

Name of the relation between LC connections in isometric immersions

Let $(\tilde{M},\tilde{g})$ be an $(n-1)$-dimensional Riemannian manifold isometrically embedded in the $n$-dimensional $(M,g)$. The L-C connections are respectively $\tilde\nabla$ and $\nabla$. $N$ ...
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1answer
75 views

Is there something like a partition of unity subordinate to something which is not a cover?

The question is basically in the title, but I‘ll explain a bit more what I mean. Let‘s say I want to construct a function with a certain property in a neighborhood of an embedded submanifold $S\...
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0answers
24 views

Orientability of manifolds with boundary

Is it true that the exterior of any knot is an orientable manifold? It is intuitive to me that an oriented manifold naturally inherits an orientation to its codimension $0$ submanifold, at least when ...
2
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0answers
115 views

Special orthogonal group is a submanifold

I work on a problem where I'm asked to proove that $SO(n)$ is a submanifold of $R^{n^2}$. If done that by defining a function $f(A)=AA^T-Id$ which satisfies the condition that $f^{-1}(0)=SO(n)$. I ...
3
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1answer
39 views

Horizontal Submanifolds of Maximal Dimension

Briefly, my question is, what algorithms or techniques exist for determining the maximal dimension of an integral submanifold of a non-integrable distribution? The Heisenberg Group $ \mathbb{H}_n $ ...
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1answer
69 views

The boundary of an open half-ball is contained in a union of two spheres ? (Lee's smooth manifold))

I found some argument in the proof of the following lemma in Lee's smooth manifold 2ed is confusing. $\textbf{Lemma 16.2}$ Suppose $U$ is an open subset of $\mathbb{R}^n$ or $\mathbb{H}^n$, and $K$ ...