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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 connected manifold

I need some tip to prove the following: If $N^{n}$ is a connected manifold and $M^{m}$ is a closed submanifold of $N$, such that $n-m\geq 2$, then $N-M$ is connected. I am supposed to use ...
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Does there exist an open neighborhood $U$ of $S$ and a smooth map $U\to S$ which is a retraction?

Suppose $M$ is a smooth manifold and $S$ a smooth embedded submanifold of $M$. Does there exist an open neighborhood $U$ of $S$ and a smooth map $U\to S$ which is a retraction?
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Intersection of two dimensional submanifolds of $\mathbb{R}^3$ with disjoint normal spaces is a one dimensional submanifold

Let $M_1, M_2$ be two two dimensional submanifolds of $\mathbb{R}^3$ such that the normal spaces $N_p$ $$N_p(M_1) \cap N_p(M_2) = \{0\}$$ for every point $p \in M_1 \cap M_2$. Then $M_1 \cap M_2$ is a ...
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How to show $A=\{(x,y)\in\Bbb R^2|x^2=y^3\}$ is not an embedded submanifold of $\Bbb R^2$? [closed]

Let $A=\{(x,y)\in\Bbb R^2|x^2=y^3\}$. How to show $A$ is not an embedded submanifold of $\Bbb R^2$? 假设$A=\{(x,y)\in\Bbb R^2|x^2=y^3\}$是$\Bbb R^2$的embedded submanifold, 根据Proposition 5.16, $(0,0)$在$...
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$SL(n,\Bbb{C})$ is a regular submanifold of $GL(n,\Bbb{C})$

Let $SL(n,\Bbb{C})$ be the group of matrices of complex entries and determinant $1$. I want to prove that $SL(n,\Bbb{C})$ is a regular submanifold of $GL(n,\Bbb{C})$. An idea is to use the ...
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If $f\in C^1$, are we able to show that $\partial\{f=0\}$ is a null set?

Let $f\in C^1(\mathbb R^d)$ for some $d\in\mathbb N$. Are we able to show that (under mild additional assumptions) $\partial\{f=0\}$ is a null set wrt the Lebesgue measure $\lambda^d$ on $\mathcal B(\...
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Does a basis for a Lie algebra of a Lie group $G$ depend on whether $G$ is embedded?

My professor recently gave the problem to find a basis for a lie algebra of a given embedded lie subgroup. The problem stressed that the lie subgroup was embedded (which was clear from the definition ...
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1answer
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Smooth images of manifolds are immersed?

in various papers in symplectic geometry, I have encountered the following argument. Statement: Suppose $f: M \rightarrow N$ is a smooth map of constant rank. Then its image $f(M)$ can be equipped ...
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1answer
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Are the matrices of nullity at most one and non-negative determinant a submanifold with boundary?

Let $\text{GL}_n^+$ be the group of real invertible $n \times n$ matrices with positive determinant. Set $S=\text{GL}_n^+ \cup \{A \, | \, \text{rank} (A) = n-1 \}$. Does $S$ form a a submanifold ...
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Computing the vector field $\partial_X Y$ (a part of an exercise regarding the torus as a minimal submanifold of $S^3$)

TLDR: How does one compute $\partial_X Y$, where $\partial$ is the Levi-Civita connection on $\mathbb{R}^n$ and $X,Y$ are vector fields on $\mathbb{R}^n$ (or possibly only defined on a submanifold of $...
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Find $f$ such that $f^{-1}(\lbrace0\rbrace)$ is a knotted curve

I would like to solve the following problem (it comes from Morris W. Hirsh, Differential Topology, it's exercise 6 section 4 chapter 1): Show that there is a $C^\infty$ map $f:D^3\to D^2$ with $0\...
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For $\Phi(x, y)=x^{2}-y^{2}$ can $\Phi^{-1}(0)$ be given a topology and a smooth str. s.t it is an immersed sub manifold?

In the book of Lee, introduction to smooth manifolds, at page 123, it is asked that \begin{array}{l}{\text { Let } \Phi : \mathbb{R}^{2} \rightarrow \mathbb{R} \text { be defined by } \Phi(x, y)=x^{...
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For which values of $a$, is ${M_{a}=\left\{(x, y) : y^{2}=x(x-1)(x-a)\right\}}$ a sub manifold?

In the book of Lee, introduction to smooth manifolds, in page 123, it is asked that \begin{array}{c}{\text { For each } a \in \mathbb{R}, \text { let } M_{a} \text { be the subset of } \mathbb{R}^{2} ...
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1answer
22 views

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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1answer
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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
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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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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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56 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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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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1answer
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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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2answers
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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
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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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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 $(e_{\geq3})^2=1$, we 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 $...
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1answer
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$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
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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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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
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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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0answers
29 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
108 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
108 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
54 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
64 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
124 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
56 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
71 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
52 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
26 views

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 $\...