Questions tagged [manifolds]
For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.
5,293 questions
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13 views
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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0answers
31 views
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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0answers
13 views
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 $...
1
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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 ...
1
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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
$$\hat{f}(\xi) = \...
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0answers
14 views
How would the chart around a point vary be different to the tangent space at the same point?
If I'm not mistaken, the tangent space and the space that a chart maps to are both Euclidean. So if I take some points from a neighbourhood around a point on the manifold, and map them to Eulclidean ...
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1answer
26 views
What is the natural isomorphism between the tangent space of a product manifold and the product of the tangent spaces?
Let $M$ and $N$ be smooth manifolds and $p\in M$, $q \in N$. What is then a natural isomorphism for
$$T_{(p,q)}(M\times N) \cong T_pM \times T_pN ?$$
3
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1answer
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 ...
3
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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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0answers
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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0answers
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 ...
3
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1answer
30 views
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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0answers
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\...
1
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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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0answers
23 views
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 ...
3
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1answer
58 views
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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0answers
14 views
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} \...
3
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0answers
49 views
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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0answers
27 views
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 ...
1
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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$ ...
4
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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 ...
1
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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$ ...
2
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0answers
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 ...
1
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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 \...
1
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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
39 views
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 ...
1
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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{...
0
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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 ...
2
votes
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 ...
3
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0answers
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
34 views
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 ...
1
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0answers
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 $M:= \lbrace (x_1,x_2,x_3) \in \mathbb R^3 : x_1\geq 0, x_1^2+x_2^2+x_3^2=1\rbrace \cup\lbrace (x_1,x_2,x_3) \in \mathbb R^3 : x_1= 0, x_1^2+x_2^2+x_3^2\leq1\rbrace$ ...
3
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0answers
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 ...
1
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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 ...
1
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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
$$B_{T_p M}(0_p,r):=\{X\in T_p M:\|X\|_g<...
1
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0answers
28 views
Taylor series on manifolds
Any analytic function $f: \mathbb{R}\to\mathbb{R}$ can be written as the Taylor series:
$$f(x) = \sum_{i=0}^\infty\frac{f^{(i)}(0)}{i!}x^i$$
I want to generalize this on manifolds.
However, if $f:...
1
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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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0answers
21 views
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 $\...
1
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0answers
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 ...
0
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0answers
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 $T$, ...
1
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0answers
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?
...
