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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.

In mathematics, a manifold of dimension $n$ is a topological space that near each point resembles $n$-dimensional Euclidean space. More precisely, each point of an $n$-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension $n$. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot.

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A question about manifolds with boundaries.

My topology textbook says the following: Let $S\subset \Bbb{R^2}$ be a closed disc. Then every point in $S$ is contained in a neighbourhood which is homeomorphic to that portion of a ball in $\Bbb{R^2}$ where $x_1,x_2\geq 0$. I don't see how this…
algebraically_speaking
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Basic Manifold Question

Below is a paragraph from the appendix from Krantz's Several Complex Variables book. I have limited knowledge regarding manifolds and was hoping (very much) that someone would be willing to provide the theorems used in each step and cite the theorem…
wellfedgremlin
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Interacting Manifolds?

Usually, manifolds have certain internal properties which are being studied on their own. However, I was wondering if a field of mathematics exists where several separate manifolds are considered together (for example overlapping) and influence the…
Kagaratsch
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How would I show if $N_1,\ldots, N_k$ are parallelizable manifolds, then so is $N_1\times \cdots \times N_k$?

A smooth $n$-manifold $N$ is called parallelizable if it admits $n$ smooth vector fields $Y_1;\ :\ :\ :\ ;\ Y_n$ that are linearly independent at every point $p$ in $M$. How would I show if if $N_1, \ldots, N_k$ are parallelizable manifolds, then so…
Kanye Brown
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Why are these meshes considered nonmanifold?

Why are these vertices and edges considered nonmanifold? What is the correct name for these surfaces(other than nonmanifold)?
pusheax
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Identifying tangent space of manifold with set

Identify $ \mathbb{R}^4$ with the space of $2×2$ matrices $M(2×2,\mathbb{R})$. The set $M$ of matrices with determinant $3$ is a smooth manifold of dimension $3$. Prove that the tangent space to M at I ( identity matrix) may be identified with the…
bytrz
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Can we always find a chart $(U,f)$ such that $f(U)=\mathbb{R}^n$

Let $M$ be an n-dimensional differentiable manifold. Can we always find a coordinate system $(U,f) $ such that $f(U)= \mathbb{R}^n$? I can see that this is indeed true for the examples I know- the sphere, finite dimensional vector space, the torus,…
gradstudent
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Equivalence of atlases

It is established that Equivalence of atlases is an equivalent relation. Now consider the real line $\mathbb{R}$ and the following one chart atlases $\mathcal{A} = \lbrace (\mathbb{R},Id)\rbrace$, $\mathcal{B} = \lbrace ( (0,1),Id^{2})\rbrace$ and…
Fred Wealthman
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Is this a topological manifold?

Consider the map $w \rightarrow (w^3, w^2)$ from $\mathbb{C}$ to $ \mathbb{C}^2$. Is the image of this map a topological manifold? I think the map is bijective and continuous, but is not a homeomorphism at $0$. So I don't know whether the image is…
WWK
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On the topology of a Riemann manifold

Given a Riemann manifold $(M,g)$,the Riemann metric induces a topology on $M$ which given by $d(p,q)$=the shortest length between $p$ and $q$,it's a metric topology,and my question is:is this topology the topology of $M$ as a manifold?If not,then…
C Weid
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Is it true that every (smooth) homomorphism of lie groups can be written a submersion followed by an immersion?

Is it true that every (smooth) homomorphism of lie groups can be written a submersion followed by an immersion? This isn't clear to me! Any help would be much appreciated!
user93826
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injective function on manifold

Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a smooth function and $M$ be a smooth manifold of $\mathbb{R}^n$. Assume that $Df(x)v \neq 0$ for all $v$ being tangent to $M$ at $x$ and for all $x$ in $M$. Can we say that $f$ is locally injective on $M$?
TaTa
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Hypersurface separating two halspaces.

I have an hypersurface separating the hyperspace in 2 halfspaces. How can I found a description (math formulas)for those two halfspaces? Thanks
Halbarad
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The manifold formed by n-dimensional positive definite symmetric matrices is of what dimension?

I know that the dimension of the manifold formed by n-dimensional symmetric matrices is $\dfrac {n(n+1)}2$.But the manifold formed by n-dimensional positive definite symmetric matrices is of what dimension?Thanks!
甄博涛
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Choosing a coordinate system to represent a curve

Let f(x,y) = 0 represent the equation of a curve in which both x and y are required to appear in the equation. Is it always possible to choose a new coordinate system such that only one (new) variable appears in the equation that describes the set…
Ahdhehshdjdj
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