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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36 views

Understanding submanifolds of $\mathbb{R}^{2 \times 2}$

The space of linear endomorphisms of $\mathbb{R}^2$, which we'll denote as $\mathbb{R}^{2 \times 2}$, is a 4-dimensional space. As such, it is not visualizable by a normal human mind. However, there ...
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1answer
23 views

How to visualize quotient manifold theorem

The quotient manifold says that if a Lie group $G$ acts smoothly, freely and properly on a smooth manifold, then the quotient space is again a smooth manifold with natural topology. All of the proofs ...
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Find a $T_pM$ and $N_pM$

I have the following Submanifold: $$M = \{(x,e^{x})\in \mathbb{R^{2}:x \in \mathbb{R}}\}$$ to which I have to find a tangent and normal space $T_{p}M$ and $N_{p}M$ at the point $p \in M$. This is one ...
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On some doubts on tangent space of immersed submanifold

In Lee's book "Introduction to Smooth manifolds", he following lemma can be found. Lemma 8.26 Let $M$ be a smooth manifold, let $S\subseteq M$ be an embedded submanifold, and let $Y$ be a ...
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1answer
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Emedding of $\mathbb{ RP}^3$

Is there a simple formula for an embedding (homeomorphic onto its image) of $\mathbb{RP}^3$ in some Euclidean space? I have seen a simple formula for $\mathbb{RP}^2$ in $\mathbb R^4$, but I can't find ...
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Differentiable structure of an $\mathbb{R}$-vector space

Let $ V $ an $ \mathbb{R} $-vector space, we can prove that $ V $ is a manifold. Using the topology compatible with $ \mathbb {R} ^ n $ and defining the chart $ (V, \phi_B) $ where $ \phi_B: V \to \...
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Question about the definition of a submanifold.

defintion (smooth manifold): A smooth manifold M is a topological manifold with an maximal atlas of compatible charts $(U_i, \psi_i)_{i \in I}$. definition (submanifold): A $n$-dimensional submanifold ...
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40 views

Condition for a submanifold to have a trivial normal bundle

Is the following assertion true? Suppose $M$ is a smooth manifold of dimension $n$, and $S$ an embedded submanifold of dimension $k$. If there is an embedding $S\times \Bbb R^{n-k}\to M$ (with $S\...
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85 views

Pair of orthogonal directions is a submanifold

I would like to ask something i don't understand. In my textbook of manifolds, it says that the subset $M$ of $\mathbb R P^{n} \times\mathbb R P^{n} $, made from the pairs $(D,D')$ of the orthogonal ...
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50 views

When is the image of $\mathbb{R}^n$ a smooth submanifold?

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be an injective $C^{k}$-function, for some positive integer $k$. Under what conditions is the complement of the open set (I know its open by Brower's ...
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39 views

On the image of an embedding of $\mathbb{R}$ in $\mathbb{R}^3$

The function $f:\mathbb{R}\to\mathbb{R}^3$ given by $t\mapsto(t,t^2,t^3)$ is clearly an injection and also an immersion. Also, using Heine Borel theorem one can show that it is a proper map, and thus ...
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Coordinate chart of a regular surface has rank 2

I'm taking a course on differential geometry and I'm using Tapp's book. There, proposition 3.21 states the following. Theorem: If $S$ is a regular surface, and $\sigma : U \subseteq \mathbb{R}^{2} \to ...
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Submanifolds are preserved under diffeomorphisms

I've been reading books about differential manifolds. There is a remark in the book that says Remark: Submanifolds are preserved under diffeomorphism, i.e If $A$ is a submanifold of $M$ and $\tau : M \...
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Restriction of a function is an immersion. Please, would you help me to improve this?

Question: Let $f, g: U \rightarrow \mathbb{R}^n$ be differentiable functions in the open set $U\subset \mathbb{R}^m$, let $\delta$ be a real positive number and $X\subset U$. Defining '$|f-g|_{1} \leq ...
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the real projective line is a submanifold of $\mathbb{RP}^{n} $

The exercise says the following: Show that a projective line is a submanifold of $\mathbb{RP}^{n}$ diffeomorphic to $\mathbb{RP}^{1}$. I think this is a consequence of the following result if $q:\...
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1answer
37 views

Show that $f^{-1}\{y\} \subseteq S^3$ is a circle for all $y \in S^2$

Let $f : S^3 → S^2$ by $f(x_0, x_1, x_2, x_3) = (x_0^2+x_1^2-x_2^2-x_3^2, 2x_0x_3+2x_1x_2, 2x_1x_3-2x_0x_2)$ Show that $f^{-1}\{y\} \subseteq S^3$ is a circle for all $y \in S^2$. Can you help me in ...
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If the invariant curves of two sub-manifolds are transversal, does this imply that the sub-manifolds are transversal?

Trying to determine the following: If the invariant curves of two sub-manifolds are transversal, does this imply that the sub-manifolds are transversal? An example I have of invariant curves, are ...
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If dim$N\geq 2$dim$M$, are Immersions are dense in $C_S^1(M,N)$?

So as the title says the question is that if dim$N\geq$dim $2M$ will immersions be dense in the strong topology? I believe this will be true , since we have that since $dim N\geq 2dim M$ we can view ...
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prove that $S^1$ is smooth submanifold of $\mathbb{R}^2$ using the definition with diffeomorphism

I'm tring to prove that the unit circle $$S^1=\{(x_1,x_2)\in\mathbb{R}^2\text{ such that }x_1^2+x_2^2=1\}$$ is an embedded submanifold of $\mathbb{R}^2$ using the following Characterization: A ...
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$X$ is a submanifold of $Y$ if and only if $X$ is a submanifold of $Z$

Let $Z$ be a differentiable manifold and $Y$ a submanifold of $Z$, let $X \subset Y$. Prove let us $X$ is a submanifold of $Y$ if and only if $X$ is a submanifold of $Z$ suppose that Z is an n-...
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$\newcommand{\R} {\mathbb R}$ Why can't an immersed submanifold of $\R^2$ have an isolated point?

I'm doing problem 5-10 in Lee's Introduction to Smooth Manifolds: $\newcommand{\R} {\mathbb R}$ Let $M_a= \{(x,y) \in \R^2 | y^2 = x(x-1)(x-a) \} \subset \R^2$ For which values of a is $M_a$ an ...
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Torus of revolution difeomorphic to $\mathbb{R}^3 \setminus \mathbb{Z}^2$

Prove by means of the function $h: \mathbb{R}^2 \to \mathbb{R}^3$ defined by $h(\theta, \varphi)=((2+ \cos \theta) \cos \varphi, (2+ \cos \theta) \sin \varphi, \sin \theta)$ that $\mathbb{R}^3 \...
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Why are retracts of connected manifolds embedded submanifolds?

I'm now working on this exercise: Let $f: M \rightarrow M$ be a smooth map on a smooth, connected manifold $M$, satisfying $f \circ f = f$, then prove that $f(M)$ is an embedded manifold of $M$. I ...
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Computation of the inner product between Laplacian of a normal vector field on the normal bundle and its normal vector field

Consider now $M^n \subset N^{n+p}$ a Riemannian submanifold. \begin{align} \Delta^{\perp}_M S = \sum\limits_{i=1}^n \nabla_{E_i}^{\perp} \nabla_{E_i}^{\perp} S - \sum\limits_{i=1}^n \nabla_{\nabla_{...
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Cover the closed ball, treated as a submanifold with boundary, with two charts

Let $$P:[0,\pi]\times[0,2\pi)\times[0,\infty)\to\mathbb R^3\;,\;\;\;(\theta,\phi,r)\mapsto r\begin{pmatrix}\sin\theta\cos\phi\\\sin\theta\sin\phi\\\cos\theta\end{pmatrix}$$ and $r\ge0$. I would like ...
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$\mathbb{S}^n \times \mathbb{S}^k$ is diffeomorphic to a submanifold of $\mathbb{R}^{n+k+1}$

Prove that $\mathbb{S}^n \times \mathbb{S}^k$ is diffeomorphic to a submanifold of $\mathbb{R}^{n+k+1}$. $\mathbb{S}^n \times \mathbb{S}^k \subset \mathbb{R}^{n+1} \times \mathbb{R}^{k+1}=\mathbb{R}^{...
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20 views

Closed embedding of the half line in a non-compact manifold

I am trying to do an exercise for some time now that is the following : Let $M$ be a connected Hausdorff non-compact paracompact $C^r$ manifold. Then there is a closed $C^r$ embedding of the half ...
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1answer
38 views

Proving this is a $2d$ submanifold

I am losing my mind over this question and am nearly convinced the question itself just is wrong. I want to show $ [(a,b,c,d) : a+b-c^3+d^2=0, a^2+b^2-8d=10]$ is a 2-submanifold of $\mathbb R^4$. I ...
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If $v(t,x)\in T_x\:\partial M$ for all $(t,x)\in[0,\tau]\times\partial M$, then $f(x)(\text{div}_{\partial M}v_0)(x)+⟨\nabla f(x),v_0(x)⟩=0$

Let $d\in\mathbb N$, $\mathcal M$ denote the set of bounded $d$-dimensional properly embedded $C^1$-submanifolds of $\mathbb R^d$ with boundary, $f\in C^1(\mathbb R^d)$, $$\mathcal G(\partial M):=\...
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Correspondence between Riemannian metrics and Euclidean embeddings

Given a sufficiently smooth manifold $M$, a Riemannian metric on $M$ induces an isometric embedding into Euclidean space by Nash's theorem, (non-canonically, non-uniquely) an embedding of $M$ into ...
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If $T_t$ is a flow and $M$ is a manifold with $T_t(\partial M)⊆\partial M$ for all $t$, we've already got $T_t(\partial M)=\partial M$ for all $t$

Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$, $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary and $f$ be a $C^1$-diffeomorphism from $\mathbb R^d$ onto $\mathbb R^d$ ...
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1answer
47 views

Proving sufficient conditions for immersed submanifolds to be embedded

Should I say anything else to prove the following from Professor Lee's Intro to Smooth Manifolds text? Thank you. Prove Proposition 5.21 (sufficient conditions for immersed submanifolds to be embedded)...
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How do we need to apply the inverse function theorem here? (diffeomorphically mapping an open subset of a submanifold)

Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$, $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$, $T$ be a $C^1$-diffeomorphism from $\mathbb R^d$ and $N:=T(M)$. It's easy to see that $...
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25 views

Showing each of the following subsets $S\subset \mathbb{C}P^n$ is a smooth manifold

Can I please have help showing each of the following subsets $S\subset \mathbb{C}P^n$ is a smooth manifold. Further, for (a) and (b), how do we identify a familiar manifold to which $S$ is ...
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1answer
25 views

Transversality and the hessian of critical ponint of a function

I am trying to do the following exercise: Let $f\in C(\mathbb{R^n,\mathbb{R}})$. Consider the gradient vector field $\nabla f=(\frac{\partial f}{\partial x_1},...,\frac{\partial f}{\partial x_n})$. ...
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1answer
50 views

Embedded submanifold and isomorphisms of the ambient space.

Pardon if this is an duplicate question. Say, for instance that I an embedded submanifold $N$ of a manifold $M$. I also know that $M$ is isomorphic (diffeomorphic?) to $M'$ via the isomorphism $f$. ...
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Show that a dashed curve is not a submanifold

I have to show that the following curve is not a submanifold. I tried to define this as a set, as the spiral logarithm, but the "dots" block me.. My definition of a submanifold is : $M$ is a ...
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1answer
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Is my proof correct? (about curve and submanifold)

I have to show that the curve on the picture is not a submanifold. Can you tell me if my proof is correct ? My works : I note $X$ the curve in the picture defining by $$ X = \left\{ (x,y) \mbox{ : } (...
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Show that a dashed spiral is not a submanifold.

The problem : it is a submanifold ? My works : I tried to write this as the following set : $$ X = \left\{ (\theta,r) \mbox{ : } r - ab^{\theta} = 0 \mbox{, } (a,b) \in \left( \mathbb{R}^{+*} \right)^...
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Let $K\subset D^{3}$. Show there exists a smooth map $f:D^{3}\rightarrow D^{2}$ such that $f^{-1}(0) = K$.

This problem comes at the end of Hirsch's $\textit{Differential Topology}$ (pg 32 #6). The picture is the following: $\hskip{2cm}$ $K$ is the knotted string. The problem asks you to show there exists ...
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1answer
51 views

Invertible Immersions are diffeomorphisms

I have been thinking about this question : Show that if $i: N\rightarrow M$ is an invertible immersion then it is a diffeomorphism. Give a counterexample when $N$ is not Second Countable. First time ...
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59 views

Grassmannian is a manifold in a specific case the $2$-planes in $\mathbb{R}^4$

I want to show that Grassmannian is a manifold in a specific case the $2$-planes in $\mathbb{R}^4$. I'm in the following context: $G(2,4)$ are the $2$-planes in $\mathbb{R}^4$ that we can identify ...
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13 views

partial integration on the boundary of a submanifold

Let $d\in\mathbb N$, $M$ be a bounded $d$-dimensional properly embedded $C^2$-submanifold$^1$ of $\mathbb R^d$ with boundary, $\nu_{\partial M}$ denote the outward-pointing unit normal field on $\...
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20 views

Contraction of tensor by metric in mean curvature flow

Picture below is from the 255th page of Huisken, Gerhard, Flow by mean curvature of convex surfaces into spheres, J. Differ. Geom. 20, 237-266 (1984). ZBL0556.53001. In my view, $$ A*\nabla A = g^{ia}...
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1answer
24 views

Existence of a coordinate system on an embedded submanifold in $\Bbb R^n$ satisfying a certain condition

Let $M$ be an embedded submanifold of dimension $k$ in $\Bbb R^n$, and let $u^1,\dots,u^k$ be coordinates for a region of $M\subset \Bbb R^n$. Then the inclusion map $M\hookrightarrow \Bbb R^n$ ...
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27 views

Why a submanifold is an open subset of its closure?

When I am reading Lie Groups and Lie Algebras I by Onishchik, I come across the claim that "As any submanifold, a Lie subgroup is an open subset of its closure." From this the author deduces ...
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1answer
18 views

if $f$ is open and has constant rank, then $f$ is a submersion.

Suppose $f:U\subseteq\mathbb{R}^m\to \mathbb{R}^{n}$ has constant rank. Show that $f$ is a submersion if and only if $f$ is open. I know that a submersion is an open function. Now if $r$ is the rank ...
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28 views

Substitution rule for the surface measure on a $C^1$-submanifold

Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$ and $\Omega$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$. Assume, for simplicity, that $\Omega$ is described by a single chart, i.e. ...
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0answers
47 views

Surjectivity of the second fundamental form

Let $M^m \subseteq N^n$ be a submanifold, with $(N, g)$ a Riemannian manifold and the Levi-civita connection denoted by $\nabla$. Consider the second fundamental form generalized to arbitrary ...
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1answer
24 views

Gram determinant of a boundary chart of a submanifold

Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$ and $\Omega$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary. Assume, for simplicity, that $\Omega$ is described by a single ...

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