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

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1answer
34 views

Integral of Differential 1-form

Let $\omega$ be the closed $1$-form $\omega = \frac{xdy - ydx}{x^2 + y^2}$ and let $S$ be the unit circle and let $C = \{(x,y) : (x-3)^2 +y^2 = 1\}$. I'm trying to find $\int _S \omega$ and $\int ...
1
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1answer
21 views

Integrate 2-Form over surface

Problem: Calculate $\int_S dx \wedge dy + dy \wedge dz$, where $S$ is the surface given by $S = \{(x,y,z) : x = z^2 +y^2 -1, x < 0\}$. Wikipedia says: Let $$ \omega=f_{z}\, \mathrm dx \wedge ...
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0answers
9 views

Sum two nearest function of two class are the nearest function of the sum class

Suppose $x,\mu:[0,1]\rightarrow \mathbb{R^2}$ two smooth function and $\Gamma = \{\gamma : [0, 1] \rightarrow [0, 1]| \gamma (0) = 0, \gamma (1) = 1, \gamma$ is a diffeomorphism $\}$. Here $\Gamma$ ...
1
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1answer
29 views

Differential of the multiplication and inverse maps on a Lie group

I'm trying to solve the following two problems: Let $G$ be a Lie group with multiplication map $\mu\colon G\times G\to G$, and let $\ell_a\colon G\to G$ and $r_b\colon G\to G$ be left and right ...
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1answer
30 views

Why does $(X_a,Y_b)(f\circ \mu(a,b)) = X_a(f\circ R_b(a)) + Y_b(f\circ L_a(b))$?

On the first page of these notes: if $\mu\colon G\times G\to G$ is the multiplication map on a Lie group $G$, then given a point $(a,b)\in G\times G$ and letting $R_b$ and $L_a$ denote ...
4
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1answer
34 views

Apparent violation of fundamental theorem of ODEs, how to resolve?

Consider, in the $(x, y)$-plane, the family of curves given by $y = (x - c)^3$, for the various possible values of the number $c$. Denote by $v$ the unit vector field everywhere tangent to this family ...
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0answers
26 views

$M=M_1\cup M_2$ is not necessarily a manifold, when $M_1\cap M_2=\emptyset$, $M_i$ a manifold

While $M_1\cap M_2=\emptyset$, $M_i$ a manifold, show $M=M_1\cup M_2$ is not necessarily a manifold. Another question, prior to this one, was to show the union is a manifold, where the conditions ...
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0answers
5 views

Vector bundle morphisms $T(I\times I)\longrightarrow A$?

Let $I:=[0, 1]$ be the unit interval in $\mathbb R$ and $\pi:A\longrightarrow M$ a vector bundle. Is there a nice characterization of the vector bundle morphisms $T(I\times I)\longrightarrow A$? ...
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2answers
30 views

Diffeomorphism group of product manifold

For a given differentiable manifold $M$, the diffeomorphism group $\mathrm{Diff}\left( M \right)$ of $M$ is the group of all $C^\infty$ diffeomorphisms of $M$ to itself. Consider a product manifold of ...
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1answer
25 views

Whats the surface area of the surface $0 \leq z, (x-1)^2 + y^2 \leq 1$?

Whats the surface area of the manifold $0 \leq z, (x-1)^2 + y^2 \leq 1$? The surface is the intersection of the sphere $x^2 + y^2 + z^2 = 4$ and a cylinder centered at $(1,0,0)$. I'm just not sure how ...
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0answers
29 views

Linearity in quotient space

Let $\mathcal{C}$ be the space of all parametric curves $x:[0,1]\rightarrow \mathbb{R}^2$. Also let $\mathcal{C}$ is a linear manifold in the sense that $x_1,x_2\in \mathcal{C}$ implies that ...
2
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1answer
25 views

Topological boundary as a submanifold

Let $U$ be an open subset of a smooth $n$-manifold. Consider $\partial U$ the topological boundary of $U$. Is the following true ? : If $\partial U$ is a smooth $n-1$ submanifold without boundary, ...
2
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1answer
84 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 $C^2$ ...
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0answers
19 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 ...
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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
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2answers
56 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
30 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 ...
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1answer
36 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
32 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 ...
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3answers
455 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 ...
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1answer
38 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 ...
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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 $g:(0,1) \times (0,\tfrac{1}{3}) \to C$ ...
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0answers
28 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
23 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 ...
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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
32 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
59 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 ...
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0answers
29 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? ...
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0answers
32 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
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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
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1answer
18 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 ...
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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 ...
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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
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1answer
27 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
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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
18 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 ...
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0answers
19 views

Is the cone of a manifold a manifold of dimension one higher?

I think the cone of a manifold in complex projective space (the preimage of it by projection) would be a manifold of dimension one higher, but I don't know how to show this.
2
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0answers
58 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) ...
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2answers
42 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 ...
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0answers
24 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
23 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
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1answer
87 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
48 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
42 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
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1answer
33 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
68 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
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1answer
46 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
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0answers
26 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 $1$?