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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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How many 2-manifolds can be covered by a single chart?

Every non empty paracompact connected 1-manifold is either homeomorphic to a circle or to the real line. Therefore, one can trivially say that all 1-manifolds (without boundary) covered by a single chart are equivalent to $\mathbb{R}$. Is there a…
Martino
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How to prove that a level set is not a submanifold

I've seen an exercise in which I am to prove that $$\{(x,y,z) \in \mathbb{R}^3 : x^2+y^2=z^2\}$$ is not a submanifold of $\mathbb{R}^3$. I've done a little bit of research and found these answers Showing that a level set is not a submanifold How to…
user116708
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Short question regarding flow on the general linear group

Say that we have a vector field on $GL(n,R)$, given by $A \rightarrow (A,A^2)$ (I mean the matrix power here). If we try to find the flow of this vector field, I get that it should be: $X(A,t) = e^{tA^2}A$, where $e^{tA^2}$ is the matrix exponential…
Shaf_math
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Differentiable manifolds, uniqueness of maximal atlases and definition of smooth manifolds maps.

I have proved that given an atlas for a topological space $M$ that a maximal atlas containing $M$ is unique. But my proof would fail to generalise to the statement that a maximal atlas conatining a chart is unique. Is this true? Also given a map $f…
joe ibbs
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A question about submanifolds

I need help with the following problem: Let $M$ be a connected smooth manifold. Let $f\colon M\to M$ be a smooth mapping satisfying $f(f(x))=f(x)$ for each $x \in M$. Show that $f(M)$ is an embedded submanifold of $M$. I'd like to prove that…
Kanae Shinjo
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One remark on Integration on Manifold

I was reading the book Introduction to Manifold by Loring W Tu. And I am confused with a remark Tu made in his book. I need a little bit of clarification. In Chapter 5 (differential forms), he wrote " Because integration of function on Euclidean…
Timon
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What does it mean that two differentiable structures are distinct?

In Loring Tu's Introduction to Manifolds, problem 6.1 (pg. 70): Let $\mathbb R$ be the real line with the differentiable structure given by the maximal atlas of the chart $(\mathbb R,\phi=\textrm{id}:\mathbb R \rightarrow \mathbb R)$, and let…
TaeNyFan
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Unfolding the $n$-dimensional sphere

Is there an extension to $n$ dimensions of the usual spherical coordinates mapping a three-dimensional sphere to a two-dimensional rectangle? [Duplicate]: Analogue of spherical coordinates in $n$-dimensions
pluton
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Is there an acyclic closed manifold (for integral homology)?

A space is acyclic if the reduced integral homology groups are all trivial. I want to know whether there exists a closed manifold which is acyclic. A necessary condition is that this manifold will not be orientable. Another necessary condition is…
Djamel
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Define a diffeomorphism for $U(m)/U(m-1)\cong S_{2m-1}$

Define a diffeomorphism for $U(m)/U(m-1)\cong S_{2m-1}$. Looking at the Differentiable Manifolds text by Shahshahani $U(m)/U(m-1)$ looks like a homeomorphism, but I'm skimming Wikipedia's text for diffeomorophisms, orbits, and quotient spaces and…
Michael T Chase
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How to tell computationally that a volume of points constitutes a manifold

Suppose that a space ${\mathbb R}^{r}$ contains a set of points which we want to consider as enclosing a volume within the space, or perhaps a volume in a submanifold (e.g., the sphere $S^{2}$ within 3-space). How do we detect computationally that…
user79774
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Cutting out submanifolds with "orthogonal" functions

Let $Z$ be an embedded manifold in some $\mathbb{R}^M$. Then locally $Z$ is cut out by independent functions $(g_1, \ldots, g_l): \mathbb{R}^M \to \mathbb{R}^l$, where $l = \operatorname{codim} Z$. That is, for every $z \in Z$, there exists a…
JHF
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Is every manifold homeomorphic to a subset of some Euclidean space?

Usual examples of manifolds that I see are (up to homeomorphism) always a subset of $\mathbb{R}^n$ (ie curves, spheres, klein bottles, etc.). Is this true for every manifold?
user736690
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Show that M is a 1-manifold in $R^2$ covered by the single coordinate patch $\alpha$

Let $\alpha : R \to R^2$ be the map$ \alpha(x)=(x,x^2)$ ;let M be the image set of $\alpha$. Show that M is a 1-manifold in $R^2$ covered by the single coordinate patch $\alpha$. I know the definition of manifold, but I am not clear how to show that…
Z.B. Zuo
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Differential forms on $M\times\mathbb{R}$

Let $\alpha$ be a differential form of degree $p+1$ on $M\times\mathbb{R}$, where $M$ is an arbitrary smooth manifold, and $p$ a non negative integer. Can $\alpha$ be always written as $\beta+\gamma\wedge dt$, where $\beta$ and $\gamma$ are…
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