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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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Why $f^{-1} (y)$ is a closed set

Let $M$ be a compact smooth manifold, $f$ is a smooth map between $M$ and $N$. If $y \in N$ is a regular value, then $f^{-1} (y)$ is a closed set. I don't know why $f^{-1} (y)$ is a closed set.
Sheldon
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Can I simplify a 3-sphere to another 3-manifold knowing a symmetry property?

First of all I have to say that I am not very competent in topology, so please try not to use too obscure terms in your answers and pardon me for my explanations that may not be very rigorous. I am using a 3-manifold that is almost (you'll see why I…
Balfar
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Example of this sort of submanifold

This is from Smooth manifolds and their applications page 7: Let $P=P^r$ be a subset of the smooth manifold $M^k$ of class $m$, defined near each of its points by a system of $k-r$ independent equations. This means that for each point $a \in P$…
snailspace
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Is a single point boundaryless?

In the question at MathSE "Is a single point boundaryless?" I saw two contrary answers both make much sense to me. One is by wendy.krieger A single point is a one-dimensional polytope, and is entirely of content, without boundary. The problem…
1LiterTears
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Orientation of manifold given by external normal field

Consider the unit sphere $S^1$ of $\mathbb{R}^2$. This is a 1-dimensional manifold. And an orientation $\sigma$ of $S^1$ is given by the orientated atlas $\left\{\phi_1,\phi_2\right\}$ with the maps $$ \phi_1\colon (-\pi,\pi)\to S^1\setminus…
user34632
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Do we really need differentiable manifolds to be modeled on R^n?

A topological manifold can be seen as a set together with a topology inherited from R^n, by the usual definition using charts. This definitions allows us to define continuous functions and limits in the set by referring to the corresponding concepts…
someone_else
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What's the point in Coordinate Functions?

A long time ago I've asked here about what O'Neill defines in his "Elementary Differential Geometry" book as "Natural Coordinate Functions". In the time, I've understood that it was a notational convenience and all of that. But now I've started to…
Gold
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How are tangent spaces related?

Suppose we have a manifold with different smooth structures. The tangent space at a point depends on the choice of maximal atlas (right?). Is there a relation between the different tangent spaces? Maybe if the topological manifold has some nice…
green day
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Does there exist a $4$-dimensional contractible open manifold that is not homeomorphic to $\Bbb{R}^4$?

This question might be answered on this platform already. However, I am lacking the expertise to make the connection myself. I read about the Mazur manifolds which are contractible manifolds with boundary which fail to be diffeomorphic (how about…
Lars D.
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Example of $\mathbb R$-points of a manifold, which is not an actual point.

A naive question inspired by the idea in algebraic geometry. Let $M$ be a smooth manifold (second countable and Hausdorff), not assuming compactness. Let $C^\infty(M)$ be the ring of smooth functions. Define an $\mathbb R$-point of $M$ as a ring…
Display Name
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$M$ is $k$ dimensional manifold connected, compact. Let $f:M \rightarrow \mathbb{R}$ smooth function. Prove there exists $x_0\in M$ that $f(x_0)=0$

Let $M$ a $k$ dimensional manifold, connected and compact. Let $f: M \rightarrow \mathbb{R}$ a smooth function. Suppose there exists $x_1,x_2\in M$ such that $f(x_1)<0
Emma
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Question on manifold : does $f(r,\theta )$ means implicitely $f(r\cos \theta ,r\sin \theta )$?

I have $\Gamma$ a manifold and $f:\Gamma \to \mathbb R$. We have in $(x,y)$ that $f(x,y)=x^2+y^2$. Now, in my lecture, they denote $(r,\theta )$ the polar coordinates. And after, they wrote : since $$\partial _r =\cos\theta \partial _x +\sin(\theta…
user841366
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Smooth mapping on $ \mathbb R$

I am going through differentiable manifolds and come across a problem: How to construct a smooth mapping $f: \mathbb R \rightarrow \mathbb R$ such that $f^{-1}(0)=0$ $f^{'}(0) \neq 0$ $\forall \epsilon >0,f^{-1}(-\epsilon ,\epsilon )$ is not…
Barricades Mysterieuses
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can $(x^3 , y)$ be a chart if $(x , y)$ is a chart on $M$?

Let $M$ be a $2$-dimensional manifold and $(U , \varphi)$ be a chart on $M$ with $\varphi = (x , y)$. Can $(U , \psi)$, with $\psi = (x^3 , y)$, be a chart on $M$? I thought no because if it were a chart, the map $\psi = g \circ \varphi : U \to…
joseabp91
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