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

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5
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1answer
69 views

Find a (simple?) counterexample to this statement about topological manifolds.

Let us assume by a topological manifold $M$ of dimension $n$ I mean a Hausdorff topological space that is locally homeomorphic to $\mathbf{R}^n$, where $n$ is fixed. I know that if $M$ is assumed ...
2
votes
1answer
34 views

How is this “Stiefel manifold”-like fiber bundle called? How to characterize it?

Let $M$ be a smooth manifold of dimension $n$. Consider the space $V_k(M)$ of pairs $(x,\alpha)$ where $x \in M$ and $\alpha$ is a linear embedding $\mathbb{R}^k \hookrightarrow T_x M$ (or ...
1
vote
0answers
18 views

What is the most accessible reference on wall-crossing?

I am looking for a nice and easy to read reference on wall-crossing (in the context of Donaldson theory). Is there some accessible reference you have to suggest? I am interested in studying Donaldson ...
1
vote
0answers
45 views

Well definedness of Lie bracket (coordinates approach)

For practice I wanted to define Lie bracet in terms of coordinates and show that this definition is independet of the choice of coordinates. So I started with the definition ...
3
votes
0answers
66 views

What is the formal definition of tangent hyperplanes?

My questions arise from 11.21 DEFINITON and 11.22 THEOREM ,both of them presented on p341 An Introduction To Analysis W.R.Wade 3ed. Question 1. Considering 11.21 DEFINITON, Let ...
1
vote
0answers
14 views

Non-degeneracy of curves/manifolds

Ok so I'm having some problems with understanding what it means for a manifold or curve to be non-degenerate. The definition I've been trying to get my head around is: "Non-degeneracy is a ...
3
votes
1answer
31 views

Cotangent bundle tensor product tangent bundle

What is the meaning of Cotangent bundle tensor product tangent bundle: $T^*M\otimes TM$? what will an element of this space be?
0
votes
1answer
34 views

degree of smooth maps from 2-sphere to 2-torus

Why any smooth map from the 2-sphere to the 2-torus has zero degree? Can we show that there is no surjective smooth map from 2-sphere to 2-torus?
-2
votes
0answers
28 views

Show a set is $k$-submanifold

Suppose you have $n-k$ differentiable functions $F_i, \ i=1, \dots, n-k$ on the space $\Bbb R^k \times \Bbb R^{n-k}$ and the system $$F_1(v_1, \dots, v_k, x_{k+1}, \dots, x_n)=0 \\ F_2(\dots)=0 \\ ...
0
votes
0answers
25 views

Submersion and some properties

Theorem: Let $f:M\to\mathbb R^m, M\subset\mathbb R^n$ be a submersion, $p\in M$ and $D_pf:\mathbb R^n\to\mathbb R^m$ is the functionalmatrix. Then there exists: - an open neighborhood $A$ on ...
3
votes
2answers
69 views

Is the pairing induced by the wedge product and integration nondegenerate on de Rham forms?

Let $M$ be a compact, oriented, smooth $n$-manifold and let $\Omega^*_{\mathrm{dR}}(M)$ be the commutative differential graded algebra of de Rham forms on $M$. We can define a pairing: \begin{align} ...
0
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0answers
20 views

Proving that the 'unit' basis vector fields for polar coordinates in the Euclidean plane are a noncoordinate basis.

This is Exercise 2.1 from Geometrical Methods of Mathematical Physics by Bernard Schutz. Show that the 'unit' basis vector fields for polar coordinates in the Euclidean plane, defined by A. $$ ...
0
votes
0answers
25 views

Comparison of orientations involving diagonals

This problem came up in a discussion about orientations and and seems more delicate than I expected: Let $M_1$, $M_2$ and $P$ be smooth oriented finite-dimensional manifolds without boundary. Let ...
1
vote
0answers
39 views

Show that $\partial(M\times N)=M\times\partial(N)$

Let M a $k$-dimensional manifold without boundary of $\mathbb{R}^{n}$ and N a $l$-dimensional manifold of $\mathbb{R}^{m}$ with or withour boundary. Show that $\partial(M\times N)=M\times\partial(N)$ ...
1
vote
0answers
49 views

Möbius strip parameterization and charts

A parameterization of the möbius-strip is given by : $$\begin{align}M=\{ (x,y,z) \in \mathbb R^3: x &= \cos t(1+ s\cos(t/2)),\\ y &= \sin t(1+ s\cos(t/2)),\\ z &= s\sin(t/2), \\ t ...
1
vote
0answers
46 views

Map from $\mathbb{R}^3 \rightarrow \mathbb{R}^6$ is Immersion for…

$$\phi :\mathbb R^3 \rightarrow \mathbb R^6$$ $$(u,v,w)\rightarrow \phi(u,v,w)=(x_1,x_2,x_3,x_4,x_5,x_6)$$ where $ \quad x_1=u^2 \quad x_2=v^2 \quad x_3=w^2 \quad x_4=vw \quad x_5=uw \quad x_2=uv$ ...
1
vote
0answers
31 views

convergence in the $C^r$-topology on $C^r(M,N)$ for $M$, $N$ compact manifolds

Let $M$ and $N$ be smooth (finite dimensional) manifolds without boundary. On the set $C^r(M,N)$ we choose the compact-open $C^r$-topology. This topology is defined as follows (I take the definition ...
6
votes
3answers
182 views

For $n$ even, antipodal map of $S^n$ is homotopic to reflection and has degree $-1$?

How do I see that for $n$ even, the antipodal map of $S^n$ is homotopic to the reflection$$r(x_1, \dots, x_{n+1}) = (-x_1, x_2, \dots, x_{n+1}),$$and therefore has degree $-1$? Thanks for your time. ...
2
votes
1answer
122 views

Why do these two facts imply that $S^n$ is not parallelizable for $n$ even, $n \ge 2$?

Consider the following two facts. If $S^n$ admits a vector field which is nowhere zero, the identity map of $S^n$ is homotopic to the antipodal map. For $n$ even, the antipodal map of $S^n$ is ...
1
vote
0answers
34 views

Is this subset of $\mathbb{R}^{3}$ a topological manifold?

Consider the set $\mathcal{M}_{1} = \{ \ (x,y,z) \in \mathbb{R}^{3}\ | \ y = -1 \ \}$, this is a plane. Also consider the set $\mathcal{M}_{2} = \{ \ (x,y,z) \in \mathbb{R}^{3}\ | \ x=y=0 \ \}$, which ...
3
votes
2answers
81 views

$S^n$ admitting nowhere zero vector field implies identity map of $S^n$ is homotopic to antipodal map? [closed]

If $S^n$ admits a vector field which is nowhere zero, does it follow that the identity map of $S^n$ is homotopic to the antipodal map?
0
votes
1answer
43 views

How does the solution of a differential equation on a manifold yield a map?

In: "A solution $x^μ(λ)$ is a map from $\mathbb{R} → M$": Why is $x^μ(λ)$ considered a map and why does it go from $\mathbb{R} → M$? I can't seem to illustrate this in my mind. In:"If the manifold ...
2
votes
1answer
46 views

Are the two standard descriptions of $\mathbb{C}P^{\infty}$ (topologically) equivalent?

While reading through some issues of Baez's (wonderful) "This Week's Finds in Mathematical Physics," I came across this statement (from week 149): $K(\mathbb{Z},2)$ is a bit more complicated: it's ...
1
vote
1answer
56 views

Prove S is a manifold.

At the moment the definition of a manifold I'm working with is that of a set $X$ equipped with a smooth atlas $A$. I want to prove that $\{(a,b)\in \mathbb{R}^n\times\mathbb{R}^n \mid a\cdot a=b \cdot ...
1
vote
1answer
20 views

Derivation of $f\in \mathcal C^1(M,N)$ where $M,N$ are smooth manifold.

I have a question about derivation of fonction $f:M\longrightarrow N$ where $M$ and $N$ are smooth manifold of dimension $n$. In my course, we try to compute $$\mathrm d_p f\left(\frac{\partial ...
0
votes
0answers
13 views

Show that the lemniscate is not a manifold. [duplicate]

Consider $\gamma(t)=(\sin(t),\sin(2t))$. How can I show that $\gamma(0)$ has no neighborhood homeomorphic to $\mathbb R$ ?
0
votes
1answer
14 views

A posteriori measures of numerical dissipation and dispersion

In PDEs, it is typical to find out how dissipative or dispersive a numerical method is by writing down the modified PDE corresponding to the numerical method, and seeing if that modified PDE contains ...
0
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0answers
16 views

Derivatives w.r.t vectors of Log map on Sphere

Let $\mathbb{S}^n$ be the unit sphere in $\mathbb{R}^{n+1}$ and $T_p\mathbb{S}^n$ be the tangent plane at $p$ of $\mathbb{S}^n$. The Log map at $p \in \mathbb{S}^n$ from $\mathbb{S}^n$ to ...
1
vote
1answer
37 views

Is the $\alpha$-curve a topological submanifold of $\mathbb{R}^2$

Consider the following subset of $\mathbb{R}^2$ defined by $\mathcal{M}=\{ \ (x,y)\in\mathbb{R}^2\ |\ y^2=x^2(x+1)\ \}$. I'm supposed to decide whether or not $\mathcal{M}$ is a topological ...
1
vote
1answer
32 views

Using the pullback to compute $f^*\mathrm d x$ where $f(r,\theta)=(x,y)=(r\cos\theta,r\sin\theta)$

My formula of the pullback is given by If $f:M\longrightarrow N$, then \begin{align*} f^*:\Omega ^k(N)&\longrightarrow \Omega ^k(M)\\ \omega &\longmapsto f^*\omega \end{align*} where ...
0
votes
0answers
18 views

Critical points and the lie bracket

Let $f\in C^{\infty}(M,\mathbb R),p\in M$ and $(x,U)$ a chart of $M$ at $p$. The points $p$ is called critical point of $f$ if $$d(f\circ x^{-1})(x(p))=0.$$ Now I want to show that for a critical ...
0
votes
0answers
53 views

Why positive definite metric is necessary to define a topology

When we define a manifold we require that it locally looks like Euclidean. But even the Lorentzian metric in SR does not locally looks like Euclidean let alone the pseudo Riemannian metric used in GR. ...
1
vote
1answer
64 views

Use of directional derivative in this proof that $\operatorname{O}(n)$ is a manifold

The following is an excerpt from a proof that the orthogonal group $\operatorname{O}(n)$ is a manifold Let $f: \mathcal{M} \simeq \mathbb{R}^{n^2}\to \mathcal{S} \simeq ...
3
votes
1answer
53 views

What is the space obtained by identifying boundary $\mathbb T^2$ of a solid torus

By identifying boundary of solid $\mathbb T^2$, one obtains a 3-manifold, but what is the space "looks like"? For example, can we understand it through Heegaard splitting? More generally I want to ask ...
0
votes
1answer
30 views

Show that a submersion is open

Let $M,N$ smooth manifold of dimension $m$ and $n$ respectively. $f:M\longrightarrow N$ a submersion. Show that $f$ is open. My proof Let $W\subset M$ an open. Let $p\in W$. By the theorem of ...
1
vote
0answers
16 views

examine if it's manifold at point $(0,0,0)$

I have problem with checking if $M=\{(x,y,z) \in \mathbb{R^3} | x^4+y^4+z^4-z^3=0 \}$ is a manifold at point $(0,0,0)$. If it is a manifold then it's function $z(x,y)$ since for $y(x,z)$ and ...
5
votes
1answer
68 views

Smooth manifold which is a group, but not a Lie Group

Are there (preferably non-pathological) examples of smooth manifolds, which are groups, but not Lie groups? In books one can see plenty of examples of Lie groups, but I haven't seen an example where ...
1
vote
0answers
24 views

Show that $\mathcal D_p(M)=\text{span}\left(\left.\frac{\partial }{\partial x^1}\right|_p,…,\left.\frac{\partial }{\partial x^n}\right|_p\right)$

Let $M$ a manifold of dimension $n$. I have to show that $$\mathcal D_p(M)=\text{span}\left(\left.\frac{\partial }{\partial x^1}\right|_p,...,\left.\frac{\partial }{\partial x^n}\right|_p\right)$$ ...
1
vote
0answers
28 views

Reducible Heegaard splittings can be written as connected sums

Recall that a Heegaard splitting is a decomposition of a three manifold into a triple $(V,W,S)$ where V and W are solid handlebodies meeting along their common boundary S. A Heegaard splitting is ...
1
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0answers
34 views

Manifolds, where its enough to have one chart for integration

Assume a compact connected manifold $M$ is given as a subset of some $\mathbb{R}^m$. Assume we have a chart $\gamma:U \rightarrow M$ such that $M-f(U)$ (the set $M$ without $f(U)$) has zero measure in ...
1
vote
1answer
57 views

Is every open subset of a manifold homeomorphic to some Euclidean space?

Let $M^n$ be a connected topological manifold. Is every proper open subset of $M$ homeomorphic to some open set in $\mathbb{R}^n$?
0
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0answers
9 views

$A=df$,check $S(q^i,r_i)=r_iq^i-f(q^i)$ is a generating function for $t_A$

the question about generating function def $S$ a generating function for the canonical transformation,if $S(q,s)$ s.t: $\Omega^*(\Theta_1-\Theta_2)=dS$,$\Theta$ be the one-form on $T^*Q$ how to ...
4
votes
1answer
53 views

Glueing manifolds with boundaries and Seifert-Van Kampen theorem

I've seen many times the following application of the SVK theorem: Let $M$ and $N$ two smooth $n$-manifolds ($n\ge 3$) with boundary and suppose that they have the same boundary $B$. Now, after ...
1
vote
2answers
56 views

Fundamental group of a manifold minus a submanifold

Let $X$ be a smooth $n$-manifold, with $n\ge 3$, such that $\pi_1(X)=\left<a_1,\ldots,a_m\right>$ (free group over $m$ elements) and suppose that there is an embedding: $$S^1\times ...
4
votes
0answers
270 views

Integral over “infinitesimal” transformed manifold

Suppose I have a $d$ dimensional manifold $\mathcal{M}$ on which I want to perform the integral of a certain function $\mathcal{f}: \mathcal{M} \longrightarrow \mathbb{R}$ I will have then ...
0
votes
0answers
45 views

Flows on manifolds

I am struggling with the following: Let $M$ be a manifold and $X$ a vector field on $M$. If $c:I\rightarrow M$ is a maximal integral curve of the vector field $X$. Then there does not exists a ...
3
votes
1answer
51 views

Integrating a density over a Mobius strip

According to this link one can integrate over a Mobius strip by using "densities". That has me very excited but I can't seem to find a reference on this. Can someone provide a book/ online source ...
2
votes
1answer
31 views

What does $P\times_G V\to B$ mean?

Let $$\pi:P\to B$$ be a principal $G$-bundle and $$\rho:G\times V\to V$$ a continuous action of $G$ on the vector space $V$. What does the notation $P\times_G V\to B$ mean? It is supposed to be ...
0
votes
1answer
20 views

$S^2$ as $3$-manifold with boundary

Consider the two dimensional sphere $S^2$. It is obviously a two dimensional topological manifold without boundary. Can one say that $S^2$ is a $3$-dimensional manifold $M$ with boundary such that ...
0
votes
0answers
48 views

The motivation of differential forms

The motivation of differential form, I think, is created to deal with the integral on manifold. But in most textbooks, differential forms is introduced by some other knowledges about covectors, tensor ...