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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Bound for the genus of the incompressible surfaces that can be embedded in a closed compact orientable irreducible 3-manifold

Let $M$ be a closed compact orientable irreducible 3-manifold. I would like to know if there exists a bound $n\in \mathbb{N}$ such that every incompressible surface embedded in $M$ has genus lesser ...
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What is the Jacobian of det???

In some questions of submanifold, I want to use determinant as a $C^{\infty}$Map, and show that, for example, $det^{-1}\{-1\}$ is a submanifold of $M_{2n}(\mathbb{R})$(this may be not true). However, ...
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Second Fundamental Form of the Graph of a Function of Higher Codimension

Let $f:\mathbb{R}^n\to\mathbb{R}^m$, $m\geq 2$, be a smooth function. I would like to find an explicit description for the second fundamental form of the graph of $f$ in terms of the Hessians and ...
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Submanifolds with codimension $0$

So pretty much as the title said, I'm supposed to find all submanifolds of $\mathbb{R}^n$ with codimension $0$. I haven't got too far but here are some of my thoughts: I started with intuitive ...
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Why is this map no an immersion at 0

I wonder why the map $f:\mathbb R\to \mathbb R^2$ defined by $f(t)=(t^2, t^3)$ is not an immersion at 0. Isn't the derivative of $f$ is $(2t, 3t^2)$ which is injective? Thank you
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Why the definition of smooth boundary defined like this and how does it imply the interior ball property?

I am trying to understand what the definition of smooth boundary is. From the following lectures notes on analysis 3 : So intuitively, a smooth boundary ( in this case , it is a $C^1$ boundary) ...
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Zeros of second fundamental form

Let $M$ be a submanifold of $N$ and suppose that $M$ and $N$ are equipped with arbitrary connections $\nabla^M$ and $\nabla^N.$ Let $A = \{ X \in TM : \mathrm{I\!I}(X,X)=0 \},$ where $\mathrm{I\!I}$ ...
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Surface with a flat umbilic point that has points on each side of the tangent plane to it

The problem is the following: Find a surface S that has a flat umbilical point P (this means $K = H = 0$ at P) such that for every open set $U \in \mathbb{R}^3, P\in U$, there are points of $S \cap U$...
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Volume of a submanifold compared to manifolds volume

Assume I have a manifold $M$ with $dim(M) = m$. The manifold is equipped with a coordinate chart $x_{i}$ such that the given metric on the manifold is: $g_{ij} = \delta_{ij} + \frac{x_{i}x_{j}}{1-\...
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About the second derivative of a symmetric function

Let $\Sigma$ be a hypesurface immersed in a Riemannian manifold $M.$ Suppose that $h$ is the second fundamental form of $\Sigma.$ Now, let $\sigma_p$ denote the $p$-th elementary symmetric polynomial ...
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Struggling to Calculate Integral of Differential Forms

The problem states this: "Let $f: \mathbb{R}^n \to \mathbb{R}$ be a $C^{\infty}$ function. Orient the graph $X = \Gamma_f$ of $f$ by requiring that the diffeomorphism $\phi: \mathbb{R}^n \to X\...
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3 votes
1 answer
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smooth maps between submanifolds, the image of tangent space under differential is contained in a tangent space

I want to show : This is from "Mathematical analysis" by Andrew Browder. This is not a manifolds text so we have only defined submanifolds on $R^n$ using the local immersion definition. I ...
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4 votes
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Higher order expansion of hypersurface about a point (beyond second fundamental form/extrinsic curvature)

Consider a smooth, compact $(d-1)$-dimensional hypersurface $S$ without boundary embedded in $\mathbb{R}^d$. The surface $S$ can be described as the graph of a function $f(x_1,x_2,\cdots,x_{d-1})$. ...
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2 votes
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What is the "derivative" of a family of submanifolds?

Suppose that $N(t)$ is a family of k-dimensional submanifolds in an n-dimensional manifold $M$, with $n > k$. Then $N(t)$ is a path in the space $\mathcal{M}$ of all k-dimensional submanifolds. I ...
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Confusion regarding the construction of a parameterization for a manifold

I am trying to prove that the differential of the inclusion map $i: X \rightarrow Y$ of a submanifold $X \subset Y$ $$di_x: T_x(X) \rightarrow T_x(Y)$$ is the inclusion map. This question has been ...
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1 answer
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Can we know that a map is a submersion with using only its level sets

Let $M$ be a manifold of dimension $2n$, $f:M\rightarrow\mathbb{R}^n$ a smooth map. We know that if $f$ is a submersion, then for any $c\in f(M)$, $f^{-1}(c)$ is a $n$-submanifold of $M$. But what ...
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Lee Smooth Manifolds, why does the Whitney Approximation Theorem fail when the co-domain has non-empty boundary?

I am trying to study chapter 6 of Lee's Introduction to Smooth Manifolds. In a remark after the Whitney Approximation Theorem, Lee stated that this theorem do not hold because it might not be possible ...
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Characterizing accelerations of paths in a submanifold

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\Hs}{\operatorname{Hess}}$ $\newcommand{\al}{\alpha}$ This is a curious inquiry: Let $f:\R^N \to \mathbb{R}^n$ be a smooth map,...
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Are these two questions asking the same thing?

Question $(I).$ Show that if the $n$-dimensional manifold $M$ is a product of spheres, then there exists an embedding $M \to \mathbb R^{n+1}.$ Question $(2).$ Show that there exists an embedding $S^{...
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2 votes
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Shape operator as a tensor field defined in a submanifold

Definition $4.18$ in O'Neill's book Semi-Riemannian geometry, with Applications to Relativity states that given a unit vector field $U$ normal to a hypersurface $M\subset \overline M$, then, there is ...
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3 votes
1 answer
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Why is any map $f:M \to \mathbb{CP}^\infty$ homotopic to a map $f_0:M \to \mathbb{CP}^1$ if $M$ is a 3-manifold?

I am reading a proof of the result that every element of the first homology a closed, oriented 3-manifold $M$ can be represented by a knot in $M$. This seems to be a pretty standard result, but I am ...
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1 answer
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How to prove that the image is a submanifold?

Here is the question I am trying to tackle: Show that the map $$f : \mathbb R P^n \to \mathbb R P^{n + 1},$$ defined by $$[p] = [p_0, \dots, p_n] \mapsto [p,0] = [p_0, \dots, p_n, 0]$$ is an embedding....
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Show that an embedding sends a conjugate transpose to transpose.

Here is the question I am trying to tackle: Prove that $$\mathbb C \to M_2(\mathbb R),$$ defined by $$x + iy \mapsto \begin{pmatrix} x & -y \\ y & x\end{pmatrix}$$ defines an embedding. Show ...
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Fake spheres and Liebmann’s theorem

In classical differential geometry, Liebmann’s theorem states that a compact and connected surface in $\mathbb{R}^{3}$ with constant Gauss curvature is a (standard) sphere; see e.g. Do Carmo's ...
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Show that there doesn't exists integral submanifold of the distribution generated by {X,Y}

This question was asked in assignment of Smooth Manifolds and I need help with the proof. Question : Let X,Y be two smooth vector fields on $\mathbb{R}^3$ given by $X= \frac{ \partial}{\partial x} + ...
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4 votes
0 answers
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Separating two non-transversal manifolds via deformation

I was reading Guillemin and Pollack's book to self-study differential topology, where I encountered the following two related problems (Exercise 5 and 6 in Section 2.3): (Exercise 5) Assume that one ...
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Implicitly defined submanifold and inclusion map

Suppose that we have a $p$-dimensional submanifold $N\subset M$ of a smooth $D$-dimensionl manifold $M$, defined implicitly by the vanishing of $(D-p)$ functions $f_\alpha(x^i)=0$, where $\alpha=1,\...
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2 votes
1 answer
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Prove the following subset of $X = \mathbb{T}^2$ is not a submanifold of $X$

I am having trouble understanding the proof of the following statement: Let $X = T^2$ be the 2-dimensional torus defined as $$T^2 := \mathbb{R}^2 / \sim $$ where $(x,y) \sim (x',y')$ iff $x-x', y-y' ...
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2 votes
1 answer
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Show that this map is an immersion but not an embbedding

Define $C=\mathbb{R}^2/\sim$, where "$\sim$" is the equivalence relation: $(x,y) \sim (x+k,y)$ and $k \in \mathbb{Z}$. Show that $\gamma:(0,\pi/3)\to C$ given by $$\gamma(t)=(1/2+ \cot(t),\...
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Restriction on submanifold and Poincare dual form

Let $N$ be a codimension $(D-p)$ submanifold of a smooth $D$-dimensional manifold $M$ and $\iota:N\to M$ be the inclusion map. Suppose that $N$ is defined implicitly by the vanishing of the scalar ...
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Is the tangent bundle of a submanifold a submanifold of the tangent bundle?

If $N\subset M$ is an immersed submanifold of a smooth manifold $M$, is its tangent bundle $TN$ an immersed submanifold of $TM$? In other words: If $\iota:N\to M$ is the immersion (smooth map with ...
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1 answer
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Why can't we apply the covariant derivative to normal vector fields?

Assume we are given an embedded Riemannian submanifold $(\mathcal{M},g)\subset (\overline{\mathcal{M}},\overline{g})$, with $\overline{\mathcal{M}}$ having the Levi-Civita connection $\nabla$ ...
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2 votes
1 answer
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Is a normal connection indeed a connection?

I am reading GTM176. In the first picture, it says that the second fundamental form is bilinear over $C^\infty(M)$, i.e., $(\tilde{\nabla}_XY)^{\bot}$ is a tensor. However, in the second picture, it ...
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1 vote
0 answers
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nonperiodic orbits of a smooth $\mathbb{R}$-action on $\mathbb{R}^2$ are embedded lines.

Let $\mathbb{R} \curvearrowright U$ be a smooth action on an open subset $U \subset \mathbb{R}^2$. Show that any non-periodic orbits under the action are embedded lines in U. My ideas so far: I have ...
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1 vote
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Subsets of a torus that are attainable from a given point by a given distribution (in the differentio-geometric sense)

Given is a $M$-dimensional torus and a known $M'$-dimensional involutive distribution $\triangle$, $M' < M$, on this torus. (Furthermore, $M'$ is known as a function of $M$.) If ${\bf x}$ is a ...
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Preimage of a submanifold under a submersion is again a submanifold

If $F:N \to M$ is a $\mathscr{C}^{k \geq 1}$-submersion and $S \subset N$ is a $\mathscr{C}^k$-submanifold, then its preimage $F^{-1}(S)$ is a $\mathscr{C}^k$ submanifold of M. My ideas so far: It is ...
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Preimage of a regular value by smooth function on bounded manifold

Suppose $f:M\to\mathbb{R}$ is a smooth function, where $M$ is a smooth, bounded manifold with boundary, $\partial M$. Suppose we knew $f_{|\partial M}=0$ and that some $c\in\mathbb{R}-\{0\}$ is a ...
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1 vote
1 answer
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Determine the conjugacy-classes in the unit-quaternions $S^3$

The question is already in the heading. Determine all conjugacy-classes of the unit-quaternions $S^3=\{h \in \mathbb{H} | \: \bar{h}\cdot h = 1\}$. My only idea to tackle this question would be: Since ...
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Understanding the definition of a regular submanifold

I have some questions about regular submanifolds as defined by Loring Tu's textbook: 1)So regular submanifolds are defined by having a coordinate neighborhood that vanishes in some coordinates. So ...
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Conjugacy groups in compact Lie groups are compact embedded submanifolds

Show that for any subgroup H < G where G is a compact Lie-group, the conjugacy groups $\{gHg^{-1} |\: g \in G\}$ form compact embedded submanifolds. My ideas: I am wondering whether "H is ...
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1 vote
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The Cartesian product of 2 embeddings is an embedding.

I am trying to solve this question: Show that the Cartesian product $f_1 \times f_2: N_1 \times N_2 \to M_1 \times M_2,$ of two embeddings $f_1, f_2$ is again an embedding. My definition of an ...
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0 answers
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Riemannian submanifold of $\mathbb{R}^3$ is uniquely characterized by it's riemannian metric?

I have an intuition which doesn't mean is correct. Suppose we have $\mathbb{R}^n$ seen as a riemannian manifold with it's standard riemannian metric (i.e. the euclidean one). Suppose we have a ...
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3 votes
0 answers
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Second fundamental form of Hypersurfaces

Consider a parametrization of a hypersurface $M \subset \mathbb{R}^n$ given by $x: U \longrightarrow \mathbb{R}^n$. It is known that the Second Fundamental Form at a point $p=x(u)$ is given by $$ II_p(...
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1 vote
2 answers
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Show that the three vector fields $X, Y$ and $Z$ on $\Bbb R^3$ are tangent to the $2$-sphere $\Bbb S^2$.

Show that the three vector fields $X = y\frac{\partial}{\partial z}-z\frac{\partial}{\partial y}, Y = z\frac{\partial}{\partial x}-x\frac{\partial}{\partial z}$ and $Z=x\frac{\partial}{\partial y}-y\...
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1 vote
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For any Cantor set $C \subset R^n$ there exists a submanifold $M \subset R^{n+1}$ such that $M \cap (R^n \times \{0\}) = C$

Show that for any Cantor set $C \subset R^n$ there exists a submanifold $M \subset R^{n+1}$ such that $M \cap (R^n \times \{0\}) = C$. I think $M$ should not be transverse to $R^n \times \{0\},$ ...
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  • 710
0 votes
1 answer
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Embedded submanifold of embedded submanifold in Euclidean space

If $S$ is an embedded submanifold in $M$ then for each $p\in S$, there exists a chart $U$ in $M$ on which $U\cap S$ is defined by the vanishing of some coordinates. My question is, does this then ...
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Is there a trick how to draw such a graph?

I'm considering the following set $$M=\{(x,y)\in \Bbb{R}^2: y(y^2-x)=0\}$$ And I want to show that this is not a submanifold of $\Bbb{R}^2$. So it is clear that the critical point is $(0,0)$ but I ...
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1 vote
0 answers
63 views

Proving that set with inequality is a submanifold

I want to show that $M=\{(x,y,z)\in\mathbb{R}^3|x-y^2=z^2+4,|x|<10\}$ is a submanifold of $\mathbb{R}^3$. The first condition is easy to fulfill with $f(x,y,z):=x-y^2-z^2-4$ and $\{f=0\}$ but how ...
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Let $M$ be a k-dim. $C^1$-Manifold of $\mathbb{R}^n$ and $\phi:\mathbb{R}^n \rightarrow \mathbb{R}^n$ a diffeomorphism, proofe $\phi(M)$ is a manifold

That task is, that given a k-dimensional $C^1$-submanifold of $\mathbb{R}^n$ and $\phi \colon \mathbb{R}^n \rightarrow\mathbb{R}^n$ a diffeomorphism. Show, that $M' := \phi(M)$ also is a $k$-...
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2 votes
1 answer
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Why do we have exactly two unit normal vectors at each point of a hypersurface in a Riemannian manifold?

I'd be interested to know why there are exactly two unit normal vectors at each point of a hypersurface in a Riemannian manifold. The book I've been using for this area of study is the one titled ...
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