# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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. ...
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 ...
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 ...