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Questions tagged [manifolds]

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

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Proving that unit quaternions are a 3 Manifold

I am very new to topology, and I am having trouble on how to prove if something is a manifold or not. The question states that: Let Q donate the set of unit quaternions (a) Show that Q is a 3-...
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Deduce the curvature of the manifold by probing it with geodesics

Suppose in order to probe the curvature of a manifold in $\Bbb R^2$, you decide to shoot off geodesics from the four corners of the square. Consider a set of geodesics, $K,$ each symmetrical about ...
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1answer
47 views

The square is not a submanifold of $\mathbb{R}^2$

Leb $X$ be the square in $\mathbb{R}^2$ $ X = \{(x,y) \in \mathbb{R}^2 : |x| + |y| = 1\} $ It's so easy to show that $X$ is a differentiable manifold of dimension one. But, it's not possible that $X$...
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Show $\int_{dM} w=\int_M dw$, use $d(w|_M)$ not $w\in \Bbb A(\Bbb R^3)$ , $w|_M \in \Bbb A(M)$.

Let $w \in \Bbb A^1(\Bbb R^3)$, $w=xzdy-yzdx$, $M=\{z=f(x^2+y^2\} over $x^2+y^2 \leq \Bbb R^2$. Illustrate Stoke's theorem with $(M,w|M)$. For easier use $f(x)=x$. Show $\int_{dM} w=\int_M dw$, use $...
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1answer
19 views

Definition of closed surface/manifold

This question might appear silly, I was reading on wikipedia that a closed surface (or manifold in general) is a surface without a boundary, I'd like to elaborate a bit on such definition. Assuming ...
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1answer
24 views

Injectivity of the Fourier transform on manifolds

Let $\mathcal M$ be a smooth compact submanifold of $\mathbb R^n$ (equipped with the standard surface area measure), and define the Fourier transform of $f \in L^2(\mathcal M)$ to be $$\hat{f}(\xi) = \...
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14 views

How would the chart around a point vary be different to the tangent space at the same point?

If I'm not mistaken, the tangent space and the space that a chart maps to are both Euclidean. So if I take some points from a neighbourhood around a point on the manifold, and map them to Eulclidean ...
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1answer
26 views

What is the natural isomorphism between the tangent space of a product manifold and the product of the tangent spaces?

Let $M$ and $N$ be smooth manifolds and $p\in M$, $q \in N$. What is then a natural isomorphism for $$T_{(p,q)}(M\times N) \cong T_pM \times T_pN ?$$
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81 views

Classification of contractible 4-manifolds

Is there a general homeomorphism classification of contractible topological 4-manifolds (possibly with boundary or noncompact)? In the compact case, any such manifold has a homology 3-sphere as its ...
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1answer
34 views

Why do we care about maximal atlases?

Let $X$ be a topological space and $\{(U_{\alpha},\phi_{\alpha})\}$ a smooth atlas. $\{(U_{\alpha},\phi_{\alpha})\}$ is maximal if for all $(U,\phi)$ as above satisfying (6), then $(U,\phi) \in \{(U_{\...
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1answer
38 views

Why do the properties of a derivation lead to a tangent space of a manifold

From these notes, https://www.dpmms.cam.ac.uk/~md384/neessnmeiwseis.pdf, definition 2.6: A derivation $D$ at $p$ is a mapping $D:X(p) \rightarrow \mathbf{R}$ satisfying $D(\lambda f+\mu g)= \lambda D ...
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87 views
+100

Identification of the tangent space of a manifold and the tangent vectors to curves

I'm studying the different definitions of the tangent space for abstract manifolds, and I'm struggling to prove that these abstract concepts reduce to the classical ones when dealing with submanifolds ...
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1answer
25 views

Quotient of Second Countable Space

I'm looking for an example for a second countable topological space $T$ such that there exist a quotient structure $T/\sim$ which is not second countable. Does there exist an example where $T$ is a ...
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1answer
50 views

Do the isomorphism classes of fiber bundles constitute a set?

Let $ M $ and $ F $ be smooth manifolds. Is the collection of isomorphism classes of fiber bundles of fiber type $ F $ over $ M $ a set or not and why?
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104 views

Does there exist a triple point map?

It is known that for every continuous map $M:S^1\to \mathbb{R}$ there are infinite double points.also by borsuk-ulam theorem this is true for each map $N:S^n\to \mathbb{R}^n, n\in \mathbb{N}$. A ...
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1answer
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Integration on manifolds?

To integrate we need a measure. A measure is a set function, $\mu$, which takes sets as arguments and spits out elements of $\mathbb{R}^*$ (positive real number with infinity included as a point). ...
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30 views

Lee Introduction to smooth manifolds problem 6-4

Need help with one of the problems in Lee's intro to smooth manifolds. The problem is as follows: (6-4) Let $M$ be a smooth manifold, and $B$ be a closed subset of $M$, and let $\delta:M\rightarrow\...
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1answer
70 views

Why are these not manifolds?

from: http://planning.cs.uiuc.edu/node132.html I don't understand visually or analytically why the three images with arrows are not manifolds. It would be nice to have an intuitive explanation too ...
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1answer
31 views

Stable and unstable manifolds that are tangent to each other in a continuous dynamical system?

I am thinking of a scenario/ examples where the stable and unstable manifold of an equilibrium of a continuous dynamical system are tangent to each other? Any examples/ plots would be helpful?
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Is my understanding of Relative Topology correct for manifolds

I am slowly encountering manifolds in my lectures, and am interested in the notion of open in relative topology with respect to manifolds: Let $M\subseteq \mathbb R^{d}$ be an $n-$dimensional ...
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1answer
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Extending $S^1$ to an embedding of $D^2$ in $\mathbb{R}^3$

Consider the specific embedding $f : S^1 \to \mathbb{R}^3$ given by, say, the unit circle in the $xy$-plane. Suppose further that this embedding is contained within a $3$-dimensional, simply-connected ...
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Derivative of d-dimensional ball yields surface of d-dimensional sphere

I want to proof the following: $B_r^d(0) = \{x\in\mathbb{R}^d\ | \|x\| < r\}$ $S_r^{d-1} = \partial B_r^d(0)$ $f:\mathbb{R}^d\to\mathbb{R}$ continous and lebesgue integrable t.s: $\frac{d}{dr} \...
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The Center Manifold of an ODE

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a vector field given by $f_i(x)=\sum_{k=1}^n \sin(x_i-x_k), \forall x\in \mathbb{R}^n $. Now consider the ODE $\dot x=f(x)$. We observe that the Jacobian of $f(x)...
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Lie algebra definition problem.

The question is: How do we define the Lie algebra generated by a set of vector fields? I'm reading Kobayashi's book Transformation Groups in Differential Geometry and in the proof of the next theorem ...
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3answers
36 views

Differentiable map between $S^2$ and $\Bbb{RP}^2$.

Let $\Phi$ be the map sending a point $(x_0,x_1,x_2)\in S^2$ to it's equivalence class $[x_0,x_1,x_0]\in \mathbb{RP}^2$. The claim is that this map is a differentiable map from $S^2$ to $\Bbb{RP}^2$ ...
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1answer
52 views

Intuition behind Riemannian-metric

I apologise in advance if something like this has been asked already and I will delete this question immediately if an already answered question of this sort clears my doubt, which is- What is a ...
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1answer
66 views

Understanding what smoothness means

Consider $\phi: A\times B \to C$, with all spaces involved topological spaces. $\phi$ is continuous if for any given neighborhood of the image point, $N_{\phi(a,b)}$, there exist neighborhoods $N_a$ ...
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77 views

Symmetric bilinear from $(d^2g)_x$ is well-defined on $\text{Ker}(dg)_x \subset T_xM$?

I'm trying to solve the following exercise on M. Audin's book $\textit{Morse Theory and Floer Homology}$ p.18. Here is the problem : Let $M$ be a smooth manifold and let $g : M \to \Bbb{R}$ be a ...
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1answer
40 views

Show the graph is a manifold

I am having trouble with showing graphs are manifolds. I would like to discuss the problem in the following specific example: Show that the following graph is a manifold. $$G_r(f) = \{(x,f(x)) \mid x \...
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1answer
43 views

Uniform distribution on Stiefel

I want to implement the method of sampling (uniformly) points on Stiefel manifold but I'm failing to find any kind of research/article/work that can give some info about the methods and techniques of ...
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1answer
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Open Cover of (0,1) by infinite collection of sets

text given in"Calculus on manifolds" In the above text, it's mentioned that no finite collection of open cover of the form $(\frac{1}{n},1-\frac{1}{n})$, n $\in N $ can cover the interval (0,1). But ...
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1answer
48 views

Smooth Manifolds: Charts or Parametrizations?

Do I have the following correct? If you define manifolds as subspaces of $\mathbb{R}^N$ as is done in Guillemin and Pollack, then you don't need charts and atlases. All you need are smooth ...
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1answer
32 views

$1$-parameter group terminology problem.

I'm reading Kobayashi's book Transformation Groups in Differential Geometry, and I'm a bit confused about the terminology that he is using at the page 3. Here is the section that I don't get: I know ...
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1answer
57 views

Existence of a special homeomorphism on $\mathbb{T}^2$.

Let $A, B$ be closed topological subspaces of $\mathbb{T}^2$. Suppose that $A$ and $B$ are homeomorphic as topological spaces. My Question: Is it possible to construct a homeomorphism $h: \mathbb{...
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1answer
32 views

Question about a proposition in Kobayashi book about $G$-structures.

I'm reading Kobayashi's book, Transformation Groups in Differential Geometry and at the page 3 is this proposition: the definition of $K$ is given here: My question is why this proposition is ...
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1answer
44 views

Boundary defining function

I'm confused about the proof of Prop 5.43 in Lee's Introduction to Smooth Manifolds Prop 5.43 Every smooth manifold with boundary admits a boundary defining function. A boundary defining function ...
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39 views

Extension of Du-Bois-Raymond lemma to Vector Fields on a Riemannian Manifold

I am trying to show the following extension of the Du Bois Raymond lemma: Let $M$ be a smooth Riemannian Manifold and $\omega: [0,1] \rightarrow M$ be a $W^{1,2}$ curve on M. Consider a tangential ...
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2answers
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Can't understand the definition of equivalence of topological atlas.

Wikipedia said The description of most manifolds requires more than one chart (a single chart is adequate for only the simplest manifolds). A specific collection of charts which covers a manifold ...
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26 views

In a 2D Riemannian manifold, how does exponential map change the distances?

How to bound from below the distance between the images of two points $x,y$ (within convexity radius) with a given distance $||x-y||$ under the exponential map in a Riemannian manifold? Let $M$, $\...
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1answer
19 views

Manifold with boundary - finding the boundary

I have the manifold with boundary $M:= \lbrace (x_1,x_2,x_3) \in \mathbb R^3 : x_1\geq 0, x_1^2+x_2^2+x_3^2=1\rbrace \cup\lbrace (x_1,x_2,x_3) \in \mathbb R^3 : x_1= 0, x_1^2+x_2^2+x_3^2\leq1\rbrace$ ...
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59 views

Geometry as a Group Action

At 38:45 in this lecture by Thurston he defines a geometry as a an action by a group $G$ on a simply connected topological space $X$ such that the action is transitive and the stabilizer of a point $x\...
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1answer
46 views

Composition of solutions to ODE on a manifold

I don't understand lemma 1.4.7 page 18 of this introduction to differential geometry (https://www.math.ens.fr/~biquard/idg2008.pdf). $\varphi_{t}\left ( {x} \right )$ is a solution to the ordinary ...
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1answer
41 views

$\exp_q^{-1}(\exp_p(tX)) = \exp_q^{-1}(p)+t\Gamma_{p\to q}(X)+O(t^2)$ as $t\to 0$?

Let $(M,g)$ be a Riemannian manifold with induced metric $d$ and injectivity radius $r>0$. Let $p, q$ be two points in $M$ such that $d(p,q)<r$. It is easy to see that $p$ and $q$ can be ...
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0answers
26 views

$\int_{T_pM}\phi\ dx = \int_M \lambda^{-d}\phi(\lambda^{-1}\exp_p^{-1}(\cdot))\ dV_g$?

Let $(M,g)$ be a Riemannian manifold, $r>0$ be its injecitivity radius, $p$ be a point on $M$. Let $\phi:T_p(M)\to \mathbf R$ be a function supported in $$B_{T_p M}(0_p,r):=\{X\in T_p M:\|X\|_g<...
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28 views

Taylor series on manifolds

Any analytic function $f: \mathbb{R}\to\mathbb{R}$ can be written as the Taylor series: $$f(x) = \sum_{i=0}^\infty\frac{f^{(i)}(0)}{i!}x^i$$ I want to generalize this on manifolds. However, if $f:...
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1answer
39 views

vector field of real projective space

Let $RP^n$ the real projective space. This is a manifold and i have take the usual charts in order to prove it. The problem is that i don't know how to define a vector field on that. Since $RP^n$ is ...
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Chart and parametrization in a manifold : a bit confusing. It look that given a chart we get a parametrization but in practice we do the opposite

I'm a bit confuse between chart and parametrization. So let $M$ a manifold (of dimension ). Let $\varphi :U\to \mathbb R^n$ a chart. If I understand well, $\varphi (m)$ is the coordinated of $m$ and $\...
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37 views

Mobius strip with constant negative curvature

Is there any simple model of the Mobius strip with a constant negative Gaussian curvature? There is an example on Wikipedia (https://en.wikipedia.org/wiki/M%C3%B6bius_strip#Open_M%C3%B6bius_band), but ...
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21 views

Under exponential map in a Riemannian manifold, what radius keeps the image of a ball a geodesic space?

Let $M$ be a Riemannian manifold, $p\in M$, $T=T_pM$, and $\exp_p:T\to M$ the exponential map. Is there a non-trivial bound from below on the maximum radius of a ball $B_r$ around the origin in $T$, ...
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44 views

Computing geodesics on pseudo-riemannian manifolds

Consider a pseudo-riemannian manifold $M$ with a metric tensor $g$. Now, given two points $p_1, p_2$ in $M$, how do I compute (as in, programatically compute) the geodesic between these two points? ...