Questions tagged [manifolds]
For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.
8,578
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Random Walk on a Discrete Circle of n-dimensions and a Differentiable Manifold of n-dimensions
Let us have a circle of k discrete points. We will walk on this 2 dimensional circle randomly with a 50%-50% distribution to go clockwise or counterclockwise. If we take the number of steps we take to ...
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Is a monomorphism of vector bundles an embedding?
Let $(p,E,B)$ and $(p',E',B)$ be two $C^r$ vector bundles over the same base space $B$. (When $r>0$ all the spaces are $C^r$ manifolds and all the maps are $C^r$ smooth. When $r=0$ they are just ...
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An example of a non hausdorff space
Here is the question I am trying to understand its solution:
Show that there exist nonorientable $1$-dimensional manifold if the Hausdorff condition is dropped from the definition of a manifold.
I ...
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Basic questions about the flat torus
I am trying to undesrtand a bit better the geometry of the flat torus $T^n=\mathbb{R}^n/{\mathbb{Z}^n}$ and during this process some question that I would like to solve have arising:
Regarding the ...
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What is the difference between a parameterization of a manifold and a local chart?
I am confused on the difference between a parameterization of a manifold $M$ and local charts on $M$. If $M$ has dimension $n$, we may find a subset $U \subset \mathbb{R}^n$ such that there exists a ...
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Closed curve in a torus diffeomorphic to a circle?
In Example 15.9 of Tu's "An Introduction to Manifolds: Second Edition", it is written
Let G be the torus $R^2/Z^2$ and L a line through the origin in R2. The torus can also be represented ...
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What is the meaning of this statement in the proof that grassmanian is a manifold?
In wiki page https://en.wikipedia.org/wiki/Grassmannian of Grassmannian, in the endowment of smooth structure to Grassmannian, I encounter this statement,
"For each ordered set of integers $1 \...
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Does a special orthogonal group $SO(4) \supset SU(2)$ contain a special unitary group? [closed]
Note that in the context of Lie group,
The spin group $Spin(4) = SU(2) \times SU(2)$ is a product of two special unitary groups.
Can you answer:
Does $SO(4)=Spin(4)/{\mathbf{Z}/2}=\frac{SU(2) \times ...
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Is the subset of symmetric $ 3\times 3 $ matrices with given eigenvalues a manifold?
For $ \mu_1>\mu_2>\mu_3 $ are three real numbers. Consider th set
$$
S(\mu_1,\mu_2,\mu_3)=\{A\in M(3,\mathbb{R}):,A^T=A,\,\,\lambda_1(A)=\mu_1,\lambda_2(A)=\mu_2,\lambda_3(A)=\mu_3\},
$$
where $ ...
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How to give an explicit manifold structure on $ \{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\} $.
Consider $ N $ defined as follows
$$
N=\{n\otimes n-m\otimes m:n,m\in S^2,\,\,(n,m)=0\}\subset M^{3\times 3},
$$
where $ S^2 $ denotes the two dimensional sphere, $ (\cdot,\cdot) $ represents the ...
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1
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Manifold of lines in $\mathbb{R}^3$
From [1, pag. 106] we learn that the set of all lines in in $\mathbb{R}^3$ is a $4$-$D$ manifold $\mathcal{S}$ embedded in $\mathbb{R}^6$:
Consider a space line $L$. Let $H$ be the point on $L$ ...
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Paraboloid and parametrizations
After having studied some concepts and definitions regarding manifolds (at a basic level), I wondered if a paraboloid with equation $P = \{ x^2 + y^2 = z\}$ was a manifold. The answer was affirmative ...
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A question on the manifold $ \{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\} $.
Consider a manifold $ N $ defined as follows
$$
N=\{n\otimes n-m\otimes m:n,m\in S^2,\,\,(n,m)=0\}\subset M^{3\times 3},
$$
where $ S^2 $ denotes the two dimensional sphere, $ (\cdot,\cdot) $ ...
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Manifold with the same dimension as $\mathbb{R}^n$
I am studying manifolds at a basic level, and I was wondering if, when a set has the same dimension as the space where it is defined (take, for instance, $\{ (x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 ...
- 1,004
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Differential forms, antisymmetry, and integrals
Is it appropriate to think of surface integrals and volume integrals that come up in physics as integrals over a region (connected subset) of a 2D and a 3D manifold, where we add up the contributions ...
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Question on equation of dimensions of tangent spaces in Proposition 5.38 of Lee's Smooth Manifolds
I am having trouble figuring out an equality on dimensions in the proof below from John Lee's Introduction to Smooth Manifolds.
In the proof, how do we get $\dim T_p M - \dim T_{\Phi(p)}N=\dim T_p S$?
...
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A regular, connected, compact surface with curvature on $[0,1]$
today was my final differential geometry exam and there was a problem that I partially solved, but I have some doubts.
The problem asked to prove that there exists a regular, connected, compact ...
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Characterization of the tangent space of an embedded submanifold via defining map
I have a confusion on function composition in the proof below by John Lee. We define $\Phi: U \to N$ to be a local defining map for $S$, i.e. $U$ is an open subset of $M$ and $S \cap U$ is a regular ...
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On the proof of the uniqueness of smooth structures on submanifolds by John Lee
I am having difficulty with an argument in the proof below. So we assume $\tilde{S}$ to be an immersed submanifold with some topology other than the subspace topology and some smooth structure, and $S$...
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$\frac{\partial }{\partial t}(f \circ \phi^{-1} \circ \phi \circ \gamma)(0)=\frac{\partial}{\partial t}(f \circ \psi^{-1} \circ\psi\circ \gamma)(0)$
I feel like this result is in a way obvious, since we can just apply the chain rule several times:
\begin{aligned}
\frac{d}{dt}(f \circ \phi^{-1} \circ \phi \circ \gamma)(0)
&= D_{f\circ\phi^{-1}\...
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Proving that the collection of PDFs using GMM will form a statistical manifold
If I have a set $X=\{x_1,x_2,...,x_n:x_{i}\in\mathbb{R^3};x_{i}\neq x_{j} \text{ for }i\neq j\}$. and suppose I use the Gaussian mixture model to map fit the probability density function to this $X$ ...
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irreducible compoenets in Hamiton's first Ricci flow paper [closed]
In "Three-manifolds with positive Ricci curvature",page 288,11.6 lemma.
I dont konw why $\left\lvert E_{ijk} \right\rvert ^2=\frac{7}{20}\left\lvert \partial_i R\right\rvert ^2$ ,I have used ...
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Optimizing linear least square subjected to manifold constraints
I am trying to optimize a function of a matrix $X \in \mathbb{R}^{m \times n}$
$$
f(X) = \frac{1}{2} \left\lVert AX - B \right\rVert_F^2
$$
s.t. $X^T X = I_n$ (identity matrix).
As I've been watching ...
- 5,263
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Fundamental group of manifold and axiom of choice
It is known that the fundamental group of a manifold must be countable.
As far as I am aware, a proof of this uses in an essential way the Axiom of Countable Choice.
I was wondering: is it consistent ...
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Notion of « piece of continuously differentiable manifold » in Lichnerowicz book
I am reading Linear algebra and analysis by André Lichnerowicz and there is a notion on which I find nothing on the internet.
First the context : We consider $E$ and $F$ real Euclidean spaces of ...
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How does the Riemannian Hessian change when changing the metric?
Let $M$ be a smooth Riemannian manifold with metric $\langle\cdot,\cdot\rangle_x$, and consider a smooth function $f:M\to\mathbb{R}$. Suppose that we consider another Riemannian metric $\langle\cdot,\...
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Degree of covering map of manidolds alternative proof
For context, I am doing Hatcher's exercise 3.3.9, which aims to show if $p: M\to N$ is a finite sheeted covering map of two closed, connected orientable manifolds, then the degree of $p$ (as defined ...
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homology of a manifold without a point
For me this appears in the context of Hatcher exercise 6.a chapter 3.3. But the question stands on its own.
Let $M$ be an $n$ dimensional closed manifold, and $p\in M$ an arbitrary point, my goal is ...
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Is there a way to convert an element of $\mathfrak{so}(3)/\mathfrak{so}(2)$ into a "geometric" tangent vector on $S^2$?
From my understanding, the tangent space at the identity of the homogeneous space $\rm SO(3) / \rm SO(2)$ is just the quotient space $\mathfrak{so}(3) / \mathfrak{so}(2)$. An element in this quotient ...
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Are $n$-dimensional manifolds locally homeomorphic to $n$-dimensional Euclidean space $\mathbb{E}^n$ or the more general real space $\mathbb{R}^n$?
Many if not most authors use the term “$n$-dimensional Euclidean space” as synonymous with “$n$-dimensional real space”, $\mathbb{R}^n$. Some, however chose to be more rigorous and use the term ...
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Can every "noodly" space be decomposed into one or more manifolds?
The Premise:
Let $X$ be a topological space. For any pair of points $x_0,x_1\in X$, let $\mathrm N_X(x_0,x_1)$ be the set of all subspaces $S$ of $X$ such that $x_0,x_1\in S$ and $S\cong I$ for some ...
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Can we obtain estimates based on local sectional/scalar/Ricci curvature knowledge? [closed]
I am interested in lower bounding the injectivity radius at a point $p$ on a complete Riemannian manifold $M$. I can find results online saying that if we have e.g. global curvature estimates, like ...
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Question about this notation for m-forms and differential forms:
In many mathematics textbooks the $m$-form on $T_p \mathbb{R}^n$ is written in a compactified notation that I haven't seen before:
$$\omega = \sum_{(1 \le i_1 < \cdots < i_m \le n)}a_{i_1 \cdots ...
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Can one prove the covering space of a manifold is second countable without using the fact that the fundamental group of a manifold is countable?
I'm trying to prove that the covering space X of a manifold Y is a manifold of the same dimension. I was stuck on proving X is second countable, more specifically, on proving the fiber of any element ...
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Change of Riemannian metric as a smooth map
I just started studying Riemannian geometry, so excuse me if I don't have all the necessary tools yet.
Let $M$ be a finite dimensional smooth Riemannian manifold and let us denote as $\langle \cdot,\...
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Does this 'n-transitivity' property of manifolds have a name?
It seems that if I take any n points in a connected manifold of 2 dimensions or more, $M$, $x_i$, and any other points in $M$, $y_i$, I can continuously move every point in $M$ around until $x_1$ is ...
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Prove that $\gamma$ is constant iff $\gamma′(t) = 0$ for all $t \in J$ if there is always an interval containing $t$ in which $ \gamma$ is const
I am trying to prove the following lemma.
Let $M$ be a smooth manifold
(a) Let $J$ be an open interval and $ \gamma: J \to M$ a smooth curve. Suppose
for every $t \in J$ there is an open interval $...
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the problem of Riemannian manifolds [closed]
(3-9) suppose G is a compact Lie group with a left-invariant metric g and a left-invariant orientation.Show that the Riemannian volume form dVg is bi-invariant.
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Manifold of $SU(2)$ and it's relation to physical rotations
I'm a physicist trying to make sense of some topics of group theory and their applications to quantum mechanics. Let me state the things I think I understand
Special Orthogonal Group $SO(3)$
This is ...
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Cohomology of product Grassmann manifolds
For infinite complex Grassmann manifolds, we always have embedding $\tau\colon G_m\times G_n\to G_{m+n}$, then how to prove the induced homomorphism of cohomology rings $\tau^*\colon \mathsf{H}^*(G_{m+...
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Tangent Bundle of a smooth manifold with boundary
This is exercise 3.19 from John Lee's Introduction to Smooth Manifolds that I can't quite prove.
Suppose $M$ is a smooth manifold with boundary. Show that $TM$ has a natural topology and smooth ...
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Pullback of a Partition of Unity
I'm trying to prove that the collection of function $\{F^*\rho_\alpha\}$ is a partition of unity on $N$ subordinate to the open cover $\{F^{-1}(U_\alpha)\}$ of $N$. Here $\{\rho_\alpha\}$ is a ...
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Proving $S^{1}$ is a manifold under the quotient topology
Lets say I have defined $S^{1}$ as $[0,1]/\sim$ where $x\sim y$ when $x=y$ except when $x,y$ equal 0 or 1. In other words,
$[0,1]/\sim$ is the singletons and the set $\{0,1\}$. I want to show that ...
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$n-$ forms and relation to Lebesgue integral [duplicate]
I am self-learning differential geometry, may I ask the following questions:
If a 1-form is a linear function $\omega :T_p\mathbb{R}^n\to \mathbb{R}$, does it mean it's just dot product of the ...
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Prove that the trajectory of the image of a 3D fixed point by a certain parametric matrix is a smooth curve in $S^2$
For $t \in \Bbb R$ we give the matrix
$R(t)=\begin{pmatrix} \cos(t) & − \sin(t) &0 \\ \sin(t)& \cos(t) & 0 \\ 0 & 0 &1\end{pmatrix}$
If $p \in \Bbb R^3$ (written as a column ...
- 3,025
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Holonomy vs structure group of tangent bundle
Given a Riemannian manifold $(M,g)$, is the holonomy group of $g$ the structure group of the oriented tangent bundle on $M$? If not, is there a relation between the two? Their definitions seem quite ...
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28
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Manifold measure as the limit of "weighted" lebesque measures.
Let $\mathcal{M} \subseteq (0,1)^n \subseteq \mathbb{R}^n$ be a d dimensional differentiable manifold and $f: [0,1]^n \rightarrow \mathbb{R}^{n-d}$ be a smooth function such that $\mathcal{M} = f^{-1}(...
- 11
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Orientability of top-dimensional manifolds (with boundary)
Is every top-dimensional manifold-with-boundary orientable? If this is not true, is there an easy-to-understand counterexample?
EDIT: Some context in case this question was unclear. The hypotheses of ...
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Is every topological submanifold in $\Bbb{R}^n$ locally a level set?
It's well known that smooth embedded submanifold in $\Bbb{R}^n$ locally a level set (and locally is a graph), as in the thread Is every embedded submanifold globally a level set? setting up.
What if ...
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Riemannian metric on fixed rank manifold
I know that one can define metrics on the manifold of SPD matrices
$$
\mathcal{S}^n = \{ A \in \mathbb{R}^{n\times n} \ | \ \text{A positive semi-definite} \}
$$
like the Log-Euclidean metric or the ...
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