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
35 views

Universial covering and fundamental group of a space of pairs

Let $M$ be the space of pairs $\{(l,P)|l \subset P \subset R^3\}$ where $l$ is a one-dimensional subspace and $P$ is a two-dimensional subspace of $R^3$. Define a injection $M \rightarrow RP^2 \times ...
2
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1answer
25 views

Is there Domain invariance for manifolds with boundary?

It is well known that for manifolds without boundary, there exist a domain invariance theorem in the following form. Theorem. A subspace in an $n$-dimensional manifold without boundary is open if and ...
0
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1answer
28 views

Not locally flat space

In physics, and particularly in general relativity, we use the notion of manifold to describe space-time. In this way we have a space that locally looks like $\mathbb{R}^n$, a "flat" space. Are there ...
1
vote
1answer
27 views

Boundary points of a manifold

I'm reading about Riemannian Geometry and my question is regarding Manifolds with Boundary. I want to show a point of a manifold with boundary is either an interior point or a boundary point, so no ...
3
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0answers
96 views
+150

How to find the integral curves that are orbits of one-parameter groups?

Consider $\mathbb{R}^2$ with standard symplectic structure and inner product. Consider a Hamiltonian $$H=(x,y)A(x,y)^t$$ where $$A=\begin{pmatrix} \alpha & \beta \\ \beta & \delta ...
0
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0answers
36 views

Is “small disk” well-defined?

I saw the notion "small disk" very frequently used in literature. For example, in Brunnian braids on surfaces by V. G. Bardakov, R. Mikhailov, V. V. Vershinin, J. Wu, one line reads: Let $P_n(M)$ ...
-5
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0answers
84 views

Mnyfld difference [closed]

what is diffrence between real manifolds and other manifolds?
-1
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0answers
28 views

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

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 ...
2
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0answers
58 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
vote
1answer
41 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
44 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
19 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
32 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
votes
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
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1answer
20 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
41 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 ...
10
votes
3answers
133 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
votes
1answer
38 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
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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
69 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
195 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
34 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
votes
0answers
26 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
18 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
vote
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
vote
1answer
62 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
30 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
vote
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
votes
2answers
69 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
10 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
vote
1answer
31 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
87 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 ...
0
votes
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
104 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
vote
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
28 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
votes
0answers
26 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
vote
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
vote
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
votes
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
votes
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
votes
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
votes
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 ...