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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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Smooth approximation (under supremum norm) of distance to algebraic set in $\mathbb{R}^n$.

Given a set $S$ which is the zeroes of a finite number of homogenous polynomials in $x\in\mathbb{R}^n$, I want a constant $\alpha$ and a $C^2$ approximation, denoted $d$, to the function $d(x,S)=\inf_{...
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Equations defining a submanifold

How do you show that some given equations define a k-dimensional submanifold in an open neighborhood of the origin? For example, I am given the equations $f_1(\mathbf{x}) = e^{x_1} + e^{x_2} + e^{...
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Inner product of fields extension on a Riemannian Manifold

Let $M$ and $\overline{M}$ be Riemannian manifolds and $f : M \rightarrow \overline{M}$ an isometric immersion. Now, consider $X, Y, Z \in \mathfrak{X}(M)$ fields defined on $M$ and let $\tilde{X}, \...
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What does continuity of the determinant say about the value of a submatrix on neighborhoods of $M_{m\times n}(\mathbb{R})$

I am following Lee's book on smooth manifolds. On pages 19 and 20 he writes the following: Suppose $m < n$, and let $D_m \subset M_{m\times n}(\mathbb{R})$ be the set of real $m\times n$ ...
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Ricci equation is trivial in codimension $1$

Let $f:M\to\overline{M}$ an isometric immersion and assume $\dim(M)=\dim(\overline{M})-1$. I'm asked to show that the Ricci equation offers no information. I guess what I have to show is that the ...
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1answer
58 views

The square is not a submanifold of $\mathbb{R}^2$

Leb $X$ be the square in $\mathbb{R}^2$ $ X = \{(x,y) \in \mathbb{R}^2 : |x| + |y| = 1\} $ It's so easy to show that $X$ is a differentiable manifold of dimension one. But, it's not possible that $X$...
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Tangent spaces of two transverse subspaces are transverse subspaces

I am very new to differential geometry and was thrown this very long question: Suppose that two subspaces $V$ and $W$ of $\mathbb{R}^n$ are transverse (so $\text{Span}(V,W)=\mathbb{R}^n$). Let $O$ be ...
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Show that all solutions are of the form $(t,t^2,t^3)$

I want to show that there exists a global map of a submanifold described as the solution of the equations below. So I found one map, but I need to know if it contains all possible solutions. How does ...
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When do two different Jacobi Fields commute in the sense of the Lie Bracket?

Let $\mathcal{M}$ an riemannian submanifold of euclidean space, i.e. $\mathcal{M} \subset \mathbb{R}^n$. Equipped with the Levi-Civita- Connection. Additionally let $\mathbf{\gamma}^1$ and $\mathbf{\...
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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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41 views

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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42 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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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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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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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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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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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
86 views

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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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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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
41 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 $\forall k$ ...
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1answer
42 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
122 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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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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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
100 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
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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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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
61 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
103 views

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
54 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
66 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
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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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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
107 views

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
63 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
105 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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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
71 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
90 views

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
125 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
132 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
171 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
106 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
48 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 ...