For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

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2
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1answer
98 views

Potential proof for the Slice-Ribbon conjecture (may be wrong).

Let $f:(D^2,S^1)\to(D^4,S^3)$ be a smooth embedding (so called a slice disk), and we set $M:=f(D^2)$. Then, is the restriction map $C^{\infty}(D^4)\to C^{\infty}(M)$ open map with relative to $C^2$ ...
1
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0answers
21 views

How to construct free subgroup of SO(3) using Baire's Theorem?

I'm looking to demonstrate that there exists a free group on two generators $ G\subseteq SO(3) $ (this is for homework, so hints are preferred to complete answers I've turned it in now, so full ...
1
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1answer
40 views

Is $\overline{u^{-1}((-\infty,a))}$ a $C^k$-manifold with boundary for a $C^k$ function $u$?

Let $u\in C^k(\mathbf{R}^d)$, $M:=\overline{u^{-1}((-\infty,a))}=\overline{\{x\in\mathbf{R}^d \mid u(x)<a\}}$, where $a$ is a constant such that $M\neq\emptyset$. Is $M$ a $C^k$-manifold with ...
3
votes
2answers
57 views

Is there a locally compact, locally connected, Hausdorff and second countable space that is “nowhere locally Euclidean”?

When I study topological manifold, I think some property of manifolds are so important that they can "almost characterize" manifolds. But I know a topological manifold is not easily to be ...
0
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0answers
32 views

Prove: The pre-image of a null-set on a manifold in $\mathbb{R}^k$ is a null-set

Prove: The pre-image of a null-set on a manifold in $\mathbb{R}^k$ is a null-set. Given a $k$ dimensional manifold, $M$, and a mapping $r: U \rightarrow M$, and a null-set $E \subset M$, prove that ...
1
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1answer
41 views

Why are are integrals of functions in coordinates on manifolds not invariant under coordinate transformations?

I'm reading the book Introduction to Smooth Manifolds. And there is a question that confuse me on page 202. Can anyone tell me why it would change under coordinate transformations graphically? Thank ...
2
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0answers
33 views

Differential of a map including a manifold

Let $f\in C^{k}(M,\mathbb R)$ with $M$ is a $m$-Manifold and $d_xf:T_xM\to\mathbb R$ is the surjectiv differential. Let $m\lt l$ and $L:\mathbb R^l\to\mathbb R^{m-1}$ be a linear map and ...
10
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3answers
464 views

What are the “technical troubles” with using a metric space rather than a topological space when defining an abstract manifold? (As in Spivak)

One thing I think is interesting about Spivak's book A Comprehensive Introduction to Differential Geometry is that Spivak uses metric spaces instead of topological spaces when defining an abstract ...
0
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1answer
39 views

calculate the surface of the manifold in $\Bbb{R}^4$

How to calculate the surface area of the following manifold : $$ x_1^2 + x_2^2 = x_3^2 + x_4^2, 0 \le x_1^2+x_2^2 \le a^2$$ I know I should first describe this manifold as a map or a graph of a ...
0
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0answers
4 views

Finding this transition function for homeomorphisms over a cylinder

Let $C = \left\{ \ (x,y,z) \in \mathbb{R}^{3} \ | \ x^{2} + y^{2} = 1,\ 0 \leq z \leq 1\ \right\}$ I've got the functions $f:(0,1) \times (0,1) \to C$ and $g:(0,1) \times (0,\tfrac{1}{3}) \to C$ ...
1
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0answers
30 views

It can happen that $M_1 \cap M_2 =\emptyset $ but $M_1 \cup M_2$ is not a $k$-dimensional manifold. Give a counter example.

Let $M_1,M_2 \in \mathbb{R}^n$ be $k$-dimensional manifolds, $ M=M_1 \cup M_2$ It can happen that $M_1 \cap M_2 =\emptyset $ but $M$ is not a $k$-dimensional manifold. Give a counter example. ...
0
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0answers
24 views

M is a k-manifold if and only if $\phi(M)$ is a k-manifold

Let $\phi: \mathbb{R}^n\rightarrow \mathbb{R}^n$ be a diffeomorphism and $M\subset \mathbb{R}^n$ M is a k-manifold if and only if $\phi(M)$ is a k-manifold. Prove it. So what I did was try to ...
1
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2answers
37 views

What do we need to guarantee that $[X, Y]_p$ is linearly independent with $X_p$ and $Y_p$?

I am trying to figure out the conditions such that $[X, Y]_p$ is linearly independent with $X_p$ and $Y_p$ for some vector fields $X, Y$ and some $p$ in a three-dimensional manifold. I have that ...
1
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0answers
17 views

How to define an atlas on this manifold with boundary?

Consider the set $\mathcal{M} = \{\ \mathbf{x} \in \mathbb{R}^{3}\ | \ 1 \leq ||\mathbf{x}|| \leq 2 \ \}$. This is a $3$-submanifold with boundary. Obviously, we have $\partial \mathcal{M} = \{\ ...
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0answers
35 views

Is the restriction of a smooth vector field to a regular submanifold also smooth?

Let $S$ be a regular submanifold of a manifold $M$, meaning a subset of $M$ such that for all $p\in S$ there is a coordinate neighborhood $(U,\phi) = (U,x^{1},...,x^{n})$ of $p$ in the maximal atlas ...
1
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1answer
62 views

Is $\mathbb{R}\times\{0,1\}$ a manifold?

The definition of a $k$-manifold we are given is a set $M\subset\mathbb{R}^n$ such that the following equivalent conditions hold for each $x\in M$: There exists a mapping ...
0
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0answers
32 views

What's the surface measure (volume) of the manifold $x_1^2 + x_2^2 = x_3^2 + x_4^2$, $0 \leq x_1^2 + x_2^2 \leq a^2$?

What's the surface measure (volume) of the manifold $x_1^2 + x_2^2 = x_3^2 + x_4^2$, $0 \leq x_1^2 + x_2^2 \leq a^2$? I'm trying to figure this out in terms of an integral of a manifold but can't ...
1
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1answer
40 views

Embedded submanifolds

Set $L= \{ (x,y)\in \mathbb{R} : x^3=y^5\} $. Consider the parametrization of the curve $ t \to (t^5,t^3) $. Then the derivate of this curve at 0 is 0. Hence $L$ is not an embedded submanifold? ...
0
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0answers
33 views

An exercise about immersion map

The map \begin{align*} F \colon \mathbb{R} \times \mathbb{C} &\to \mathbb{C}^2 \\ (t,z) & \mapsto (z^2,tz) \end{align*} restricts to an immersion $f \colon S^2 \to \mathbb{C}^2$, where $ S^2 ...
1
vote
1answer
12 views

Extending covering projection of the boundary

Let $M$ and $E$ be (topological) manifolds with boundaries $\partial M$ and $\partial E$ respectively and assume we have a finite-sheeted covering $\rho: \partial E\to \partial M$. Is it possible to ...
0
votes
1answer
19 views

Jacobian of a diffeomorphism

Let $U,V\subseteq \mathbb{R}^{n}$ be open. Let $\alpha:U \to V$ be a smooth homeomorphism. Furthermore, assume that $\mathcal{J}_{\alpha}(\mathbf{x})$ (the Jacobian matrix) has rank $n$ for all ...
1
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1answer
31 views

Orient Manifold

$\mathbf{Problem \,2.}$ Consider the $2$-manifold in $\Bbb R^3$ given by $$x^2+y^2+z^2=1,\qquad z\ge 0.$$ Orient $M$ such that $\alpha$ in the Equation $(2)$ belongs to the orientation, and give ...
1
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2answers
26 views

Tangent Space Well Defined?

Question: Let $M$ be a $k$-manifold of class $C^r$ in $\mathbb R^n$. Let $p\in M$. Show that the tangent space to $M$ at $p$ is well-defined, independent of choice patch. Unsure if I'm ...
0
votes
1answer
28 views

Quotient space of a linear space space is also linear?

Suppose, $C$ is a linear manifold (i.e., manifold which is closed under addition and multiplication) and $\Gamma$ is a Lie group. Can we say in general $C/\Gamma$ is also a linear manifold? Can we ...
2
votes
1answer
42 views

Vector field on manifold

I've only seen a vector field $V$ on a manifold $M$ as a mapping $V:M\to TM$. Is it true that they can also be seen as a mapping $V:C^{\infty}\left(M\right)\to C^{\infty}\left(M\right)$? How would $V$ ...
0
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0answers
19 views

Proving that every $n$-submanifold of $\mathbb{R}^{n}$ has a natural orientation

Let's say that $\mathcal{M}$ is a smooth submanifold of dimension $n$, of $\mathbb{R}^{n}$. Using my definition, this means: for every point $p \in \mathcal{M}$ there exists a coordinate patch ...
2
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0answers
61 views

Smooth vs topological orientation

I am confused about the different notions for a closed smooth manifold to be oriented. In my mind there are several equivalent ones: 1) Coherent pointwise orientation of the tangent spaces. 2) ...
1
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2answers
43 views

If $\int_M \omega=0\Rightarrow \omega=d\varphi$, then $H^n_c(M)\simeq\mathbb{R}$? ($M$ is a connected orientable manifold)

I'm reading a book in wich the author uses this argumet the whole time. For example, he assumes that $\int_\mathbb{R}\omega=0$ then $\omega =df$ and then he concludes that that ...
3
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0answers
30 views

Do the level sets of a constant rank map give a foliation of the domain?

I'm working through Smooth Manifolds by Lee and came to problem 19-8, where you're asked to prove that the level sets of a submersion form a foliation of the domain. However when I try to solve this ...
3
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0answers
25 views

Describing the Unit Normal to a Cylinder in $\mathbb{R}^{3}$

This is problem 34.4 in Munkres' Analysis on Manifolds. Let $\mathcal{C} = \{ \ (x,y,z) \in \mathbb{R}^{3}\mid x^{2} + y^{2} = 1 \text{ and } 0 \leq z \leq 1 \ \}$. Orient $\mathcal{C}$ by declaring ...
0
votes
1answer
90 views

I have no idea what Differential Forms are… [closed]

So in my Calc 3 class we use Shifrin's "Multivariable Mathematics", and his discussion on Differential Forms and Integration on Manifolds is impossible for me to follow. Can someone recommend ...
2
votes
1answer
49 views

Formal proof that (x,|x|) is not a smooth submanifold of $\mathbb{R}^2$

I have perused the related questions on this site, and was unable to find a formal proof of the fact stated in the title. Essentially, I have two questions: Is it a fact that if $M$ is a ...
3
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1answer
44 views

Foliations vs Laminations

What's the big difference/similarity between foliations and laminations? What kind of theorems hold for both of them? Is there something that makes them essentially the same/different?
2
votes
1answer
37 views

Are there countably many closed manifolds in each dimension?

There is a single closed topological 1-manifold (up to, of course, homeomorphism): $S^1$. The classification of surfaces shows that there are countably many closed topological 2-manifolds. ...
9
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0answers
73 views

Is $T(M)$ necessarily equivalent to the normal bundle of $M$ in $\textbf{R}^{2n}$, if $M$ is totally real?

Consider real submanifolds, $M^n \subset \textbf{C}^n$. ($\textbf{C}^n \equiv (\textbf{R}^{2n}, J)$, where $J(x, y) = (-y, x)$). $M^n$ is totally real if $J(T_pM) \cap T_pM = 0$, for all $p \in M$. ...
7
votes
1answer
47 views

Embedding of $2$-torus minus one point into $\mathbb{R}^2$ as an open set?

Let $M^2$ be the $2$-torus minus one point. Is there an embedding of $M^2$ into $\mathbb{R}^2$ as an open set?
3
votes
0answers
27 views

Can $S^k \times S^n$ be embedded into $\textbf{R}^{k + n + 1}$? [duplicate]

As the question title suggests, can $S^k \times S^n$ be embedded into $\textbf{R}^{k + n + 1}$, i.e. in codimension $1$?
8
votes
1answer
51 views

Immersion of $M^n$ into $\mathbb{R}^n$, is $M^n$ orientable? Compact? [closed]

Say we have an immersion of $M^n$ into $\mathbb{R}^n$ (same dimension). I have two questions. Is $M^n$ orientable? Is $M^n$ compact? Thanks in advance!
3
votes
0answers
22 views

On the meaning of formal sums of $k$-cubes, i.e. $k$-chains (in integration on manifolds)

A singular $k$-cube in $A \subseteq \mathbb R^n$ is a continuous function $c : [0,1]^k \to A$. A singular $0$-cube in $A$ is then a function $f : \{0\}\to A$, what amounts to the same thing, a point ...
1
vote
1answer
69 views

Where to find about the category theoretic study of manifolds?

I'm looking for a resource about a category theoretic study of manifolds. What do you think is a good start? Hint: Not after very advanced resources. So no worries (indeed, preferred) if it's an ...
1
vote
1answer
38 views

What's the normal space of the manifold $z = x^2 + y^2$?

What's the normal space of the manifold $z = x^2 + y^2$? Let's say I have a continous function $g: M \rightarrow S^n$ that sends every point on the manifold to the unit normal vector. I want to know ...
0
votes
1answer
53 views

Show the parametrized torus is a 2-dimensional smooth submanifold of$\mathbb{R}^3$ [duplicate]

How can I show that the parametrized torus $T=\{(x,y,z)\in \mathbb{R}^3 : (\sqrt{x^2 +y^2}-a)^2 +z^2 =b^2 \}$ is a 2-dimensional smooth submanifold of $\mathbb{R}^3$ ? I was thinking of using the ...
1
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2answers
39 views

Why is $f(U) \cap V$ the zero set of $y^{n+1}, \dots, y^m$? [duplicate]

I'm studying Tu's proof (p. 123) of theorem 11.13, but I just have a question about one detail. He has that $f\colon N\to M$ is an embedding of a manifold of dimension $n$ in a manifold of dimension ...
0
votes
2answers
77 views

Let $f : A\subset \mathbb{R}^{n+1} \to \mathbb{R}$, what does mean that $f$ is a submersion?

I am trying to answer the following question: Let $M_a := \{ (x^1,\ldots,x^n,x^{n+1}) \in \mathbb{R}^{n+1} : (x^1)^2 + \cdots +(x^n)^2 - (x^{n+1})^2 = a\}$. For which values of $a$, $M_a$ is a ...
2
votes
1answer
36 views

Computing Sectional Curvature on Hyperbolic Plane

For a pair of $(X,Y)$ of linearly independent vectors in $T_pM$, $p\in M$, the sectional curvature is defined as $$K_p(X,Y)=\frac{<R(X,Y)Y,X>}{|X|^2 |Y|^2 - <X,Y>^2}$$ The problem I'm ...
3
votes
1answer
42 views

Equivalent definition of properly discontinuous action

In the book An Introduction to Differentiable Manifolds and Riemannian Geometry by Boothby in Chapter $3$ the author gives the following definition: Definition($8.1$) A discrete group $\Gamma$ ...
0
votes
0answers
16 views

Given a smooth curve denoted by $(f(u),0,(g(u))$, if the rotation of the curve around the $z$-axis is a manifold, is $g$ one-to-one? Is it onto?

Given a smooth curve denoted by $(f(u),0,(g(u))$, if the rotation of the curve around the $z$-axis is a smooth manifold, is $g$ one-to-one? Is it onto? By rotation, I mean $(u,v) \rightarrow ...
2
votes
1answer
85 views

Prove: $h \in T_xM \iff \operatorname{dist}(x+\epsilon h,M) = o(\epsilon)$

Prove: $h \in T_xM \iff \operatorname{dist}(x+\epsilon h,M) = o(\epsilon)$ For $M$ a smooth manifold in $\mathbb{R}^n$, $h\in\mathbb{R}^n$, $x \in M$. I know that $T_xM = \operatorname{Im}(Df) = ...
0
votes
1answer
20 views

Uniqueness of dimension of regular submanifold

Suppose $N$ is a manifold of dimension $n$. Now a regular submanifold $S$ of $N$ of dimension $k$ is defined as, if for every point $p$ of $S$ there is a coordinate chart $(U,u_*)$ from a maximal ...
1
vote
1answer
31 views

Proof for showing that a set of space curves form a manifold

I basically have a smooth space curve $\alpha$,with curvature $\kappa$ and $\tau$, both non-zero, and I generate a family of curves $M_{\alpha} = \{\dfrac{\alpha}{\mu} : \mu \in (0, \infty) \}$ . The ...