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

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0answers
26 views

Mnyfld difference [on hold]

تفاوت بین منیفلد های حقیقی و منیفلد های دیگر
-2
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0answers
20 views

a local maximum of a $C^∞$ function $f:M→R$ is a critical point of $f$ [on hold]

A real-valued function $f:M→R$ on a manifold is said to have a local maximum at $p∈M$ if there is a neighborhood $U$ of $p$ such that $f(p)≥f(q)$ for all $q∈U. $ a) We know if a differentiable ...
1
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0answers
40 views

At what conditions a compact metric space of covering dimension $n$ (on $\mathbb R^n$) is an n-manifold?

In the discussion to the MSE post an answer of @MatthewPancia with correction in a comment of @JasonDeVito would state: Every compact metric space of covering dimension $n$ can be embedded ...
1
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1answer
35 views

A geodesic metric space is a manifold on its own right. What are conditions for a Finsler space to be a manifold?

A geodesic metric space can locally be approximated by a vector space. This approximation provides it with a natural manifold structure. It means that geodesic metric space is more fundamental concept ...
2
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1answer
41 views

Differentiable manifolds that allow isometric transition maps.

What is the class of differentiable n-dimensional manifolds that allow a differential structure, in which all transition maps are isometric? Note that isometric must be overlapping pieces of charts ...
-1
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0answers
18 views

Orientable manifold [duplicate]

I need help for this question: Let $M$, $N$ manifolds, $M$ orientable and $f: M \longrightarrow N$ local diffeomorphism, then $N$ too is orientable. I was trying by definition of orientable ...
1
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0answers
17 views

Extrinsic curvature tensors

I risk of sounding too vague, but I am interested if there are other tensors reflecting the extrinsic geometry of a submanifold other than the second fundamental form? The first fundamental form ...
0
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0answers
26 views

Completion of hyperbolic 3-manifold (Thurston's lecture note)

I need help with a part of Thurston's lecture note. p.55~56 in the 4th lecture note states that when obtaining hyperbolic manifold by gluing polyhedra having some ideal vertices, the set of ...
0
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1answer
43 views

Are open sets and open balls the same thing?

I am trying to solve the first exercise in John Lee's Introduction to Smooth Manifolds and I am confused by the terminology in the question. He says (paraphrased): Consider the usual definition of ...
0
votes
1answer
19 views

something about diffeomorphism

Suppose $A$ and $B$ are both open sets, and there is a diffeomorphism $g$ between them. My book says that the chain rule implies that $Dg$ is non-singular. I don't understand. Can anyone tell my why?
2
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0answers
40 views

Confusion about domains in PDE and “every $C^k$ manifold can be thought of as a smooth manifold”

I have read that every $C^k$ domain can be described instead by $C^\infty$ chart maps. So in this sense every $C^k$ domain is a $C^\infty$ domain. But consider standard PDE where people say thing like ...
8
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2answers
99 views

Is every self-homeomorphism homotopic to a diffeomorphism?

Given a smooth manifold $M$, is every homeomorphism $M \to M$ homotopic to a diffeomorphism? Hirsch's "Differential Topology" has a proof that every $C^1$ diffeomorphism of smooth manifolds is ...
0
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1answer
36 views

$C^l$ diffeomorphism between a smooth manifold and a $C^k$ manifold

Let $M$ and $N$ be two Riemannian manifolds. $M$ is smooth while $N$ is $C^k$ manifold. Suppose there is a $C^l$ diffeomorphism between the two manifolds for $l \leq k$. Is it true that $N$ is also ...
0
votes
0answers
35 views

Showing the sphere is not homeomorphic to a torus (my own question!) (or indeed a circle and a washer) - OR puncturing is not continuous

Motivation imagine a rubber sheet extended over the end of a tube, I am saying: "there is no continuous transformation that can retract this sheet over the side" - it is common place to talk about ...
6
votes
1answer
67 views

Is the Whitney embedding theorem tight for all $n$?

Whitney's embedding theorem states that any smooth $n$-manifold $M$ can be smoothly embedded into $\Bbb R^{2n}$. In dimension 1 this is tight: the circle cannot be embedded into $\Bbb R^1$. It is a ...
7
votes
3answers
189 views

“Drawable” Examples of Vector Bundles

I'm looking for examples of vector bundles that can be easily drawn or "illustrated" on a whiteboard for a talk I am giving. I know of a couple simple examples that I could use: When our base ...
1
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1answer
33 views

Find an example of n-dimensional differentiable manifold

Find an example of $n$-dimensional differentiable manifold whose points are not points of the variety $\mathbb{R}^{n}$
2
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0answers
25 views

Approximation of $L^2$ function by smooth functions on a manifold

Let $M$ be a $C^2$ compact Riemannian manifold with boundary. Suppose $f \in L^2(M)$ is such that $0 \leq f \leq 1$. Is it possible to find $f_n \in C^\infty(M)$ such that $0 \leq f_n \leq 1$ for ...
0
votes
1answer
17 views

Interior of a compact manifold with boundary is compact

In the context of manifold with boundary, closed manifold, compact manifold I have the following question in my mind : Let $M$ be a compact manifold with non-empty boundary $\partial M$. Then ...
1
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0answers
34 views

Heisenberg group as a suspension

I'm working on Heisenberg group and I want to understand the suspension viewpoint. Let me be more precise. Let us denote by $\mathbb{H}^3(A)$ the set of matrix \begin{equation} \begin{pmatrix} 1 ...
1
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1answer
61 views

Manifold which is union of two balls is topologically a sphere

In Petersen's book while proving sphere theorem the following fact has been stated without prove : Let $M$ be a connected $n$ dimensional smooth manifold such that $M=B_{1}\cup B_{2}$ where $B_{i}$'s ...
1
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1answer
29 views

Covering space action on an orientable manifold $M$ implies $M/G$ orientable (Hatcher)

I'm trying to solve the following problem from Hatcher (3.3.4) Given a covering space action of a group $G$ on an orientable manifold $M$ by orientation preserving homeomorphisms, show that $M/G$ is ...
0
votes
1answer
24 views

Curvature line parametrization

I have a question about the curvature line parametrization. We said that for a given surface $f: U \rightarrow \mathbb{R}^3$ we find a local curvature line parametrization such that both the first ...
2
votes
1answer
39 views

Smooth function on intersection is the difference of two smooth functions

I am trying to understand a proof from Loring W. Tu's An introduction to Manifolds. In order to prove Proposition 26.2, The author must show that if $\{U, V\}$ is a open cover of a manifold $M$ and ...
1
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1answer
26 views

Finding a chart around a point in a submanifold

Define the set $M:= \{ x \in \mathbb R^4 : x_1x_4 - x_2x_3 =1 \}$, so $M$ is a 3-dimensional submanifold of $\mathbb R^4$. I want to find a chart around $a=(a_1,a_2,a_3,a_4) \in M$, but I don't have ...
3
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2answers
68 views

Show $H_2(M, \mathbb{Z}) = \mathbb{Z^r}$ if $M$ is orientable, $\mathbb{Z^{r-1}} \oplus \mathbb{Z_2}$ if nonorientable

I'm trying to solve this problem from Hatcher 3.3.24. Let $M$ be a closed connected 3-manifold, and write $H_1(M, \mathbb{Z})$ as $\mathbb{Z^r} \oplus T$ where $T$ is torsion. Show that $H_2(M, ...
0
votes
0answers
9 views

Mapping the sphere with different maps question

I am reading a physics textbook on manifolds. I am reading that in the sphere we can introduce two patches that their union covers the whole sphere. Ok, I understand why we need at least two. The maps ...
1
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1answer
30 views

Show that for a degree 1 map $f: M \rightarrow N$ the induced map $f_*: H_1(M) \rightarrow H_1(N)$ is a surjection

I'm trying to solve the following problem: Show that for a degree 1 map $f: M \rightarrow N$ of connected, closed, orientable manifolds, the induced map $f_*: \pi_1(M) \rightarrow \pi_1(N)$ is ...
5
votes
1answer
86 views

Cancellation in topological product

I was wondering whether $M\times \mathbb{R}$ is homeomorphic to $N\times \mathbb{R}$ implies $M$ is homeomorphic to $N$, where let us say $M,N$ are smooth manifolds. (They are certainly homotopy ...
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0answers
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Manifolds and Random Number Generators [closed]

I was reading this answer on quora: http://www.quora.com/What-are-the-most-important-uses-for-randomness/answer/Subit-Chakrabarti and was wondering about the following passage: Of course, a much ...
0
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2answers
49 views

How to show the covering space of an orientable manifold is orientable

I'm trying to prove this using purely topological arguments, no differential geometry as I haven't been exposed to it. I've been playing around with definitions a bit and here's what I have so far. ...
2
votes
3answers
101 views

$M \times N$ orientable if and only if $M, N$ orientable

For two manifolds $M$ and $N$ I'm trying to prove that $M \times N$ is orientable if and only if $M$ and $N$ are orientable. My attempt so far: $\impliedby)$ Assume $M, N$ are orientable. Then ...
1
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1answer
38 views

What is the difference between intrinsic and extrinsic manifold?

I'm asking this question because a course change on differential geometry at my university has updated the wording from extrinsic manifold to intrinsic manifold. This got me wonder as to what the ...
1
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0answers
24 views

How to count the number of closed manifolds in an alpha-shapes triangulion?

I have used CGAL to construct the alpha-shapes around a set of particles representing a droplet. Therefore, I have a list of surface nodes and elements forming the triangular surface. When I visualize ...
0
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0answers
24 views

How to show $g^{-1}\circ f:\partial N\longrightarrow \partial N$ extends to a diffeomorphism $h:N\longrightarrow N$?

Let $M$ and $N$ be two manifolds with boundary and let $$f, g:\partial N\longrightarrow \partial M,$$ two isotopic diffeomorphisms, that is, there exists a diffeomorphism $$F:\partial N\times [0, ...
1
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0answers
35 views

What is the formula for $\frac{\partial}{\partial x_j}(f \circ F)$?

Let $F=(F_i)_{i=1}^n: X \to Y$ be a map between two manifolds. Suppose that $(U, x_1, \ldots, x_n)$ is a local coordinate on $X$ and $(V, y_1, \ldots, y_m)$ is a local coordinate on $Y$. Suppose that ...
1
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0answers
44 views

Prove that a $C^\infty$ vector field on $M$ can be extended to a $C^\infty$ vector field on $N$ .

Suppose a $C^\infty$ manifold $M$ is a closed regular submanifold of $N$. Prove that a $C^\infty$ vector field on $M$ can be extended to a $C^\infty$ vector field on $N$ . I have no idea how to create ...
0
votes
1answer
55 views

Milnor's proof of the fundamental theorem of algebra (Topology from the Differentiable Viewpoint)

I am studying the proof of the fundamental theorem of algebra out of John Milnor's book Topology from the Differentiable Viewpoint, located on page 8 here: ...
2
votes
1answer
18 views

How to show $(d\pi^{-1})_{\pi(y)}\circ (d\pi)_x:T_xS^n\longrightarrow T_y S^n$ reverses orientation for $n$ even?

Let $\mathbb R\mathbb P^n$ be the quotient manifold $S^{n}/R$ where $R$ is the equivalence relation given by: $$xRy\Leftrightarrow y=x\ \textrm{or}\ y=-x.$$ We know the canonical map ...
0
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0answers
16 views

Possibly notation problems involving Integration and pullbacks on k-forms

$^*$ means the pullback of a k-form in this example. I cannot see how the underlined expressions have been found 1) I think that $(c \circ G)^*\omega = G^*(c^* \omega)$ but I cannot see why $c^* ...
0
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0answers
32 views

Topological boundry on orientable manifold

Let $X\subset \mathbb R^n$ be a non-empty $(n-1)$-dimensional sub-manifold for some $n\geq 2$. Assume there exists some open $U\subset\mathbb R^n$ with $x\subset U$ and a differentiable function ...
2
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2answers
25 views

rank of function on connected manifold

Let $X$ be a connected $n$-dimensional manifold and $f:X\to X$ a differentiable function satisfying: $f\circ f =f$. Prove that for all $p\in X$ that $rk_pf\leq rk_{f(p)}f$ and subsequently that $rk ...
1
vote
1answer
25 views

What are the charts that make up an atlas for the long line?

This question is prompted while I was working through the MIT OCW (Massachusetts Institute of Technology, Open CourseWare) for 18.965, ``Geometry of Manifolds,'' in its Lecture 2, ...
0
votes
1answer
35 views

Prove $O(n)$ is compact

I have to prove $O(n)$ is compact, I know if I can prove it bounded and closed in $\mathbb{R^{n\times n}}$, I will be done. But how to check boundedness and closed ness. For closedness I would like to ...
0
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0answers
25 views

Converting a word problem to algebra

This is a forming of an equation, which I haven't been able to get my head around. I have a worked solution to this problem. Question: For $x\in\mathbb{R}^m$ and $\epsilon>0$, show that ...
1
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0answers
31 views

Line with two origins is a manifold but not Hausdorff

The line with two origins is $(\mathbb{R} \times \{0,1\})/\sim$ where $(x,0)\sim(x,1)$ for $x\neq 0$. I can see that it is not Hausdorff, since we cannot separate the points $(0,0)$ and $(0,1)$. ...
0
votes
1answer
32 views

Local coordinates on a product of two manifolds.

Let $X, Y$ be two manifolds. Let $(U, x_1, \ldots, x_n)$ and $(V, y_1, \ldots, y_m)$ local coordinates of $X, Y$ respectively. I think that a local coordinate on $X \times Y$ is $(U \times V, x_1 ...
1
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1answer
27 views

A Smooth map homotopic to a constant map

Q: Let $M^{k}$ be a smooth compact $k$-manifold and let $F:M \rightarrow S^{n}$ be a smooth map, where $n>k$. Prove that $F$ is homotopic to a constant map. Proof: Since $n>k$, by Sard's ...
1
vote
1answer
33 views

Topological structure of the Manifold valued functions

$M$ is a Riemannian manifold. What condition on $M$ for $\mathcal{C}_{[a,b]}(M)$ (the set of continuous functions of the real interval $I=[a,b]$ to $M$) to be a polish space ? For which topology ? Is ...
1
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0answers
16 views

Proof of Triangulation Theorem for 1-Manifolds

While I am reading "Introduction to Topological Manifolds" by John M. Lee, I come to see the following paragraph in the proof of Theorem 5.10 pp. 102. Note that Int$\ e\cap\ $Int$\ e'$ is open in ...