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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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Prove that every embedded submanifold is naturally a manifold

it's my first question, I know it's easy but I just dont know how to write this demonstration, any help is welcome, thanks!
Deathxax
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The set of self-adjoint maps $P$ with $P^2=P$ and rank $k$ is a submanifold

Let $G_{k,m}$ be the set of the self-adjoint linear maps $P:\mathbb R^m \rightarrow \mathbb R^m$ of rank $k$ satisfying $P\circ P = P.$ Prove that $G_{k,m}$ is a $C^\infty$ submanifold of dimension $k(m-k)$ in $\mathbb R^{m^2}$. Since $P$, being…
user2345678
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Application of Manifolds to Psychology?

First paragraph of this page, asserts that the theory of mathematical manifolds find application in psychology: They appear "[i]n psychology as spaces of sensations (for example, colours)". An inconclusive search led me to believe that the author…
テレビ スクリーン
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Is there a "non-curved" manifold?

I've seen some visualizations of manifolds. It seems that they are all "curved" shapes. Is there a "non-curved" manifold?
niebayes
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My question is about manifold related to submersion and immersion

Let $N$ and $M$ be a manifolds of respectively dimensions $n$ and $m$. If a smooth map ( $M$ from $N$ )is an immersion at a point $p$ in $N$ then it has constant rank $n$ in a neighborhood of $p$. If a smooth map is a submersion at a point $p$ in…
Aera
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A proof that is too easy to be right, of the Whitney Embedding Theorem

The Whitney Embedding Theorem says that Any smooth compact $m$ dimensional manifold $M$ can be smoothly embedded in $R^{2m+1}$. Bredon's "Topology and Geometry" says that the proof of this is beyond our scope. I seem to have found an extremely…
user67803
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Charts in manifold

I have a minor problem on understanding the definition. Say, in the book An Introduction of Manifolds by Loring Tu, he gave a definition as below. My question is: are the chart $(U,\phi)$ and $(V,\psi)$ mentioned in the definition belongs to the…
Eric
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A submanifold of $\mathbb{R^2}$ (help with the rank)

Define $M=\{(x,y) \in \mathbb{R^2}\space|\space x^y=y^x , x>0, y> 0,(x,y)\neq(e,e)\}$. Show that $M$ is a one-dimensional submanifold. Here's what I have done so far. Define $$A=\{(x,y) \in \mathbb{R^2}\space|\ x>0, y>…
Célio Augusto
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Showing that a subset is a submanifold?

I'm new in the subject of manifolds, and I do not know very much about them: I just think about it as subsets of some n-dimensional Eculidean space, that are smooth etc. But I don't know/cannot use the rigorous definitions of manifolds at the…
Applied mathematician
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Submanifold that is closed

Suppose $N$ is an immersed submanifold of a smooth manifold $M$, such that $N$ has the subspace topology, and is a closed set in $M$. Let $n = \mathrm{dim}\ N$ and $m = \mathrm{dim}\ M$. Given any $x \in N$, the problem is to show that there exists…
Rolanda Hooch
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Cover of Shifrin Multivariable Mathematics with Manifolds?

Just curious, anyone knows what this image on the cover of Shifrin's textbook is? It doesn't seem to be a manifold.
Eric Miller
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Showing $P^2 (\mathbf R) $ is homeomorphic

I am trying to show $P^2 (\mathbf R) $ is homeomorphic to $D^2/\sim$ where $D^2$ is the disc and it is quotiented by the relation $(r,\phi) \sim (r', \phi')$ if and only if either $(r,\phi) = (r', \phi')$ or $r=r'=1$ and $\phi = \phi' + \pi (\mod 2…
goobie
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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 a set of lines that passes through the point…
jane
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How can I prove that "If $M$ is contractible differentiable manifold, then $M$ is orientable?"

If $M$ is a contractible differentiable manifold, then $M$ is orientable.
user58930
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Show that preimage is an embedded surface

I have given a function $$f:\mathbb{R}^4 \to \mathbb{R}^3: (x,y,z,u) \mapsto (xz-y^2, yu-z^2,xu-yz)$$ and I want to show that $f^{-1}(0)\setminus\{0\}$ is an embedded surface. If it would be an embedded curve I could verify that $f$ is an submersion…
user516068
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