For questions about smooth manifolds.

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0answers
5 views

Using partition of unit to extend paths on vector bundles in time dependent sections.

Let $\pi: A\longrightarrow M$ be a vector bundle and $a: I\longrightarrow A$ be a path where $I=[0, 1]$. I would like to show there are time dependent section $\alpha: I\times M\longrightarrow A$ ...
11
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0answers
87 views

Orientability of $m\times n$ matrices with rank $r$

I know that $$M_{m,n,r} = \{ A \in {\rm Mat}(m \times n,\Bbb R) \mid {\rm rank}(A)= r\}$$is a submanifold of $\Bbb R^{mn}$ of codimension $(m-r)(n-r)$. For example, we have that $M_{2, 3, 1}$ is ...
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0answers
21 views

Orientation form on manifold cut out by $m$ functions

Let $f: \mathbb{R}^n \to \mathbb{R}^m$ be a smooth function. If $a$ is such that $f$ has surjective derivative at all points in $f^{-1}(a)$ then this is an $n-m$ dimensional manifold $X$. I'm trying ...
2
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1answer
50 views

Understanding algebraic operations on vector bundles

Let $X$ be a smooth manifold with vector bundles $V$ and $W$. I'm trying to understand how we can construct in general a vector bundle $V \otimes W$. In particular what manifold structure do we give ...
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0answers
30 views

Condition for set of de Rham cohomology classes is linearly independent

If we have a set of 1-forms $w_1, ... w_n$ on a smooth manifold $X$ I can show that $w_i$ are linearly dependent if and only if $w_1 \wedge ... \wedge w_n = 0$. I wondered if this is also true in the ...
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0answers
20 views

Boundary of a sublevel

Let $f: B_1(0)\subset\mathbb{R}^2 \to \mathbb{R}$ be a smooth function, and say $c\in \mathbb{R}$ is not a critical value of $f$. Is it true that each connected component $\Gamma$ of $\left\{f\le ...
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1answer
22 views

Construct vector field along a curve

Let $M$ be a smooth manifold. I am trying to construct a (piecewise smooth) vector field $V$ along a curve $c$ that takes on prescribed values $K_{i}$ at times $t_{i}$. Say we construct V along ...
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2answers
47 views

Is the Lie derivative $L_{X}(\omega \wedge \mu)$ an exact form?

Let $\omega$ be an $n$-form and $\mu$ be an $m$-form where both are acting on a manifold $M$. Is the Lie derivative $L_{X}(\omega \wedge \mu)$ where $X$ is a smooth vector field acting on $M$ an ...
3
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2answers
24 views

Can a Submanifold Become Tangent to a Nowhere Tangent Vector Field

$\newcommand{\R}{\mathbf R}$ Let $M=\R^2$ and $S=\{(0, t):-1<t < -1\}$ be a submanifold of $M$. Let $V$ be a vector field on $M$ which is nowhere tangent to $S$. Let $\theta$ be the flow of ...
2
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3answers
109 views

What are interesting corollaries of a manifold being parallellizable?

This is a heavily edited (in fact, a complete rewrite) of a question I asked badly a few days ago. I am editing as opposed to asking a new question as there are already several relevant answers. I ...
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1answer
39 views

Show that a smooth manifold modulo diffeomorphism group is a smooth manifold

Would like help in starting this exercise: Suppose $\Gamma$ is a group of diffeomorphisms of a manifold $\left( {X,C_X^\infty } \right)$. Suppose that the action of $\Gamma$ is ...
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0answers
18 views

constructing a manifold structure for a cylinder [closed]

Any help on this problem would be greatly appreciated. thanks! let M be the cylinder {$(x,y,z)\in \mathbb{R}^3:x^2+y^2=1$} in $\mathbb{R}^3$. Construct a manifold structure each topological space ...
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1answer
21 views

constructing a manifold structure for a plane in $\mathbb{R}^3$ [closed]

Any help on this problem would be greatly appreciated. thanks! Let M be the plane in $\mathbb{R}^3$ with normal vector (a,b,c)$\neq$0. Construct a manifold structure each topological space (M,$\tau$) ...
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0answers
20 views

Real Analitic Manifolds, Tubular Neighborhood, Radius of Convergence

Given a Real Analytic Manifold isometrically embedded into an Euclidean Space. Gicven the maximum value of the radius of a Tubular Neighborhood "around" the manifold: what relation does it have with ...
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0answers
26 views

Injectivity Radius vs. Radius of Convergence in Analytic Manifolds

I would like to ask the following: How does the Injectivity Radius relate to the Radius of Convergence (of the analytic function to its power series) of any local (parametrization) map in the ...
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2answers
24 views

Whether the tangent space can be saw as $M\times R^m$?

Let $M$ be a smooth differential n-dim manifold, $TM$ is the tangent space ,I think the $TM$ can be treated as $M\times R^m$ ,whether it is right ?
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16 views

Principal bundle isomorphism.

Let $G\longrightarrow P\overset{\pi}{\longrightarrow} M$ be a differentiable principal bundle, i.e. $M$ and $P$ are differentiable manifolds, $G$ is a Lie group, $\pi$ is a differentiable surjective ...
-2
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1answer
22 views

Convergence pw if converges in Lp space

Let $p\in[1,\infty]$ be given. If $f$ and $g$ are non-negative analytic functions such that the following holds: \begin{equation} ...
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0answers
22 views

Converting a vector $v \in \mathbb{R}^2$ given in Polar coordinates to Cartesian coordinates

I know that switching inbetween Polar coordinates and Cartesian coordinates in $\mathbb{R}^2$ can, on suitable open subsets of $\mathbb{R}^2$, be done via $(x, y) = (r cos \theta, r sin \theta)$. Let ...
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1answer
35 views

proper action on homogeneous space

Let $M = G/K$ be a homogeneous space. It is easy to show, that the left action of $G$ on itself by multiplication is a free and proper action. My question is, if the induced action $$G \times G/K ...
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2answers
46 views

Proving Euler's spiral is an isometric embedding with bounded image

$\newcommand{\al}{\alpha}$ I am trying to prove Euler's spiral is an isometric embedding of $\mathbb{R}$ into $\mathbb{R}^2$ with bounded image. Here is the definition of the spiral: $(*) \, ...
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0answers
25 views

Minimal requirements to be a submersion.

I saw here (A surjective map which is not a submersion) that a smooth differentiable map $f:M\to N$ between two manifolds $M$ and $N$ is not necessarily a submersion. A counterexample is ...
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0answers
26 views

Showing that Killing vector fields form a vector space without introducing connection

I'm trying to show that the sum of two killing vector fields is again a vector field without introducing a connection and just using the smooth structure on a manifold. Let $X,Y$ be Killing vector ...
0
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1answer
33 views

Prove there exists a smooth unit normal at the boundary of the following manifold

Let $M$ be a compact subset of $\mathbb{R}^3$ with smooth boundary $S=\partial M$. Consider M with the standard orientation $\mu=\mu_{0}$ from $\mathbb{R}^3$ and $S$ with the boundary orientation ...
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0answers
17 views

Intuition of Immersed versus Embedded Submanifolds

The definitions I read in Lee's Smooth Manifolds is: Embedded Submanifold: $S\subset M$ is an embedded submanifold if $S \to M$ is an embedding. Immersed Submanifold: $S\subset M$ is an immersed ...
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2answers
32 views

Maps between manifolds with boundary and homeomorphism

Assume we have $f:(M,\partial M)\rightarrow (N,\partial N)$ connected 3-manifolds, not compact, such that $f$ is an homeomorphism onto its image and $f(\partial M)=\partial N$. Can say that $f$ has to ...
3
votes
1answer
190 views

How can we define $\partial x_{i_r}^p(X_p^r)$?

Let $M$ be a smooth manifold and $X_r^s:M\to T_r^s(M)$ is a section. Let $P\subseteq M$ be an open set and $X_1,\ldots,X_s\in\mathcal T(P)$ and $X^1,\ldots,X^r\in\mathcal T^*(P)$ where $\mathcal T(P)$ ...
2
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1answer
26 views

Is an open subset $U \subset \mathbf{R}^{n}$ diffeomorphic to the product $U' \times \mathbf{R}$ with $U' \subset \mathbf{R}^{n - 1}$ open?

I'm trying to prove that $U$ is diffeomorphic to the product of some open subset $U' \subset \mathbf{R}^{n}$ with $\mathbf{R}$, $U' \times \mathbf{R}$. I received the hint that this set admits a ...
3
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2answers
92 views

Trying to prove that $TM$ is a manifold: Is this function an homeomorphism?

I am trying to prove that if $M$ is a $k$-manifold in $\mathbb R^n$, then $TM=\{(p, v): p \in M, v \in T_pM\}$ is a manifold. Here, $T_pM$ is defined as a subset of $\mathbb R^n$. I know that ...
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0answers
30 views

Smooth coverings are open maps proof verification

Let $M,N$ be connected, smooth manifolds. A function $F:M \rightarrow N$ is a local diffeomorphism if for all $p \in M$ there exists open $U \subseteq M$ with $p \in U$ such that $F(U) \subseteq N$ is ...
3
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0answers
58 views

What are the essential tools and proof techniques for beginning smooth manifolds and differential topology?

I am an undergraduate currently taking a first course in smooth manifolds. I feel that I understand the material intuitively. But, I'm having trouble turning my intuition into proofs. I was hoping ...
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2answers
30 views

Given $p \in S^{n-1}$, how does one show that the map $\pi: SO_n \to S^{n-1}$ by $A \to A(p)$ is a submersion?

Pick $ p \in S^{n-1} \subset \mathbb{R}^n$ and consider the map $\pi: SO_n \to S^{n-1}$ by $A \to A(p)$. Show that this map is a smooth submersion. For $ q \in S^{n-1}$, describe the pre-image. For ...
3
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1answer
32 views

How many examples exist of Lie groups that are 2-dimesional surfaces?

It is relatively easy to show that $\mathbb{R}^2$ or $\mathbb{T}^2$ are 2-dimensional surfaces with a structure of Lie groups. I can not find other surface which are also a Lie group, there are more ...
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1answer
23 views

Does parallelizability by a frame of commuting vector fields imply those fields are coordinate vector fields of a global chart?

If $M$ is a parallelizable manifold, is the following true? If $(X_i)$ is a (global) frame of $M$ and $[X_i,X_j]=0$ for all $i,j$, then there is a global chart for which $X_i=\partial_i$.
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2answers
46 views

The orbit of a compact Lie group action

Let $G$ be a compact Lie group acting on a manifold $M$. For each $p\in M$, we define the orbit of $p$ as $G\cdot p:=\{g\cdot p: g\in G\}$. The isotropy group of $p$ is $G_{p}=\{g\in G:g\cdot p = ...
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1answer
65 views

Tangent bundle of sphere as a complex manifold

I'm trying to show that the tangent bundle, $TS^n$ of the n-sphere $S^n$ is diffeomorphic to the set $\sum z_i^2 = 1$ in $\mathbb{C}^{n+1}$. It's relatively straightforward to see that the tangent ...
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0answers
22 views

Normal vector field for an immersion

I know that a hypersurface $M$ of a riemannian manifold $N$ is orientable iff there exists a globally defined unit normal vector field $\eta : M \to TM^{\perp} \subset TN|_M$. Does the same hold for ...
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2answers
46 views

Formula of a two form for a parallelizable manifold

Let $M^n$ be a parallelizable manifold with the nowhere dependent vector fields $X_1,\ldots, X_n$ forming a basis for the tangent space at each point of $M$. The Lie brackets of these fields are ...
2
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1answer
37 views

Show that a k-form can be expressed as wedge product

I'm trying to show that given a $1$-form $\omega$ and a $k$-form $\alpha$ such that $\alpha \wedge \omega = 0$ then there exists a $(k-1)$-form $\beta$ such that $\alpha = \omega \wedge \beta$. I'm ...
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0answers
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Easier way to induce an orientation to the border of a manifold

I'm working in the following exercise: Let $M=\{(x, y, z): x²+y²=1 \,\text{and}\, 0\leq z \leq 1\}$. Let $\alpha:(0,1)²\rightarrow \mathbb R³$ be given by $\alpha(u, v)=(\cos u, \sin u, v)$. ...
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2answers
248 views

The quotient of a manifold by a submanifold is never a manifold?

Let $M$ be a connected smooth manifold. Let $S$ be a connected embedded submanifold of positive dimension and co-dimension, which is also a closed subset of $M$. Is it true that the quotient space ...
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1answer
29 views

Showing that the “abstract” tangent space of a submanifold of the $\mathbb{R}^d$ is isomorphic to the tangent space that's a subset of $\mathbb{R}^n$

Let $M$ be an $n$-dimensional smooth submanifold of the $\mathbb{R}^d$, and $p \in M$. Let $T_p^{A}M$ denote the "abstract" tangent space of $M$ in a point $p$, given by $T_p^AM = \{\gamma: ...
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0answers
25 views

How is “one of the coordinates non negative” for points in the unit circle?

Looking at a problem, Consider $S^1 \subset \mathbb{R}$. Define $U_{a,b}$ where the indexing set $I=\{(a,b):a\in\{1,-1\},b \in \{1,2\}\}$ is given, so that $U_{a,b}=\{(x_1,x_2) \in S^1:ax_b ...
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1answer
7 views

Understanding the Chow-Rashevsky Theorem

I'm trying to understand the Chow-Rashevsky Theorem. I unfortunately do not have a formal knowledge of what's going on but have figured out most of the terms. Basically a system $\Sigma$ must ...
2
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1answer
28 views

Level set of the hopf map

The Hopf map is given by the projection $\pi: \mathbb{S}^3 \to \mathbb{S}^2$, and: $\pi: z \mapsto zi\bar{z}$, where $z \in \mathbb{S}^3$ and $i \in \mathbb{H}$ is a unit quaternion. Show that ...
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3answers
67 views

Degree 1 map from torus to sphere

I'm trying to find a smooth degree 1 map from the torus $T^2 = S^1 \times S^1$ to the 2-sphere $S^2$. My first thought was to use the two coordinates $(\theta_1,\theta_2)$ to map onto the usual ...
0
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1answer
20 views

$S^2 \times S^2$ diffeomorphic to oriented $2$ dimensional vector subspaces of $\mathbb{R}^2$? [duplicate]

As the question title says, is the product of spheres $S^2 \times S^2$ diffeomorphic to the set of oriented $2$ dimensional vector subspaces of $\mathbb{R}^2$?
1
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1answer
36 views

Intersection of kernels of linearly independent smooth 1-forms on $\mathbb R^n$

I'm trying to solve the following problem: Let $\omega^1,\dots,\omega^k$ be smooth $1$-forms on $\mathbb R^n$ that are linearly independent at each point of $\mathbb R^n$. For $p\in\mathbb R^n$, ...
0
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0answers
32 views

Orientability of the level set of a map between abstract oriented manifold

Let M and N be oriented manifold and let $f:M\to N$ be a smooth map between them. Suppose $y \in N$ is a regular value for $f$, how can we show that $f^{-1}(y)$ is orientable? I've seen a solution ...
3
votes
1answer
29 views

Example of a free group action that is not proper.

I have been trying to think about Lie group actions on smooth manifolds and what the quotient spaces look like. I have a proof that compact Lie groups produce proper actions on manifolds, as well as ...