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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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Examples of closed manifolds?

In Spivak's Diff Geom (vol.1), p.19, he says a closed manifold is non-bounded and compact (A point in boundary has a neighborhood homeomorphic to half-space). I don't know a non-trivial example of that. For example, compact subset of $\mathbb{R}^2$…
Charlie Chang
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Which mathematical criteria can I use to check either a surface is homeomorphic to a disk?

I'm currently working on 3D mesh segmentation. And I want the segmented parts obtained to be homeomorphic to a disk (I mean easily unfoldable, so that we can map it as a 2D part without any points overlaping). I have read a lot of papers on the…
Gatsu
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Explicitly giving an atlas to make $SL(2, \mathbb{R})$ a smooth manifold.

In my manifolds class, we have the following theorem at our disposal: Let $Z \subseteq \mathbb{R}^n$ and suppose for each $p \in Z$ there is a neighborhood $N \subseteq \mathbb{R}^n$ of $p$ and a $C^k$ function $f: N \rightarrow \mathbb{R}$ such…
Wyatt Gregory
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Sufficient conditions - Convergence on manifold

I have came to certain problem in a differential question, but I fail to solve it. I look for sufficient conditions to make $q$ to converge to zero. \begin{equation} \alpha \dot{q} + \lambda q = 0 \end{equation} $\alpha$, $\lambda$ $\in…
Bruno Lobo
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paste Torus to itself

Suppose $Y$ is a topological space were obtained by pasting solid Torus to itself via the boundary map $F:S^1\times S^1\to S^1\times S^1$, $F(z,w)=(z^aw^b,z^cw^d)$ where $a,b,c,d\in\mathbb{Z}$ and $ad-bc=1$. I) Show that $F$ is a diffeomorphism. II)…
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The confusion usage of atlas and maximal atlas

I'm studying Loring Tu's An Introduction to Manifolds. In p.60 he said Given a smooth manifold $M=(\underline{M},\Phi_{\text{maxi}})$, it is understood by people that there exist a maximal atlas $\Phi_{\text{maxi}})$ of that underlying set…
Eric
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Help showing homeomorphism

In another question I asked for help to proving the two different definitions of projective planes are homeomorphic. I tried to write the details in proof then realised I can't finish the proof. I need help for the last step, please. What I have…
goobie
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I'm having trouble understanding manifolds.

I'm studying calculus and our professor gave us two definitions of manifolds: $1) M\subset{R^n}$ is a k-dimensional manifold if for every $x\in M$ there exists a neighborhood $U_x\subset R^n$ such that $M\cap U_x$ is a graph, i.e there exist…
Tamir Shalev
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The graph of $f:\mathbb R^2\to \mathbb R$ can be embbeded in $\mathbb R^2$ or only in $\mathbb R^3$?

Let $f:\mathbb R^2\to \mathbb R$ a continuous function. I'm a bit confuse with something : The manifold is $$M=\{(x,y,z)\mid z=f(x,y)\},$$ so it's a subset of $\mathbb R^3$. This mean that $M$ is a submanifold of $\mathbb R^3$. A parametrization of…
Dylan
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Projective space is a manifold

I have no one else to correct my work since I am not going to school therefore I'd be very grateful if someone check my work. I was trying to show that the projective real plane is a manifold: On $D^2 = \{(r,\varphi) \mid r \le 1\}$ define the…
goobie
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3-manifold embedded in 1-connected 4-manifold

Assume that 3-manifold $M$ is embedded in 1-connected 4-manifold $N$ and it separates it onto two manifolds $N_1,N_2$ with boundary. In such case from Mayer-Vietoris we have: $H_2N\to H_1M\to H_1N_1\oplus H_1N_2 \to H_1N $ If $N$ is also 2-connected…
mmm
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Gluing cylinders together

I was trying to glue two cylinders and then show that the resulting space is a manifold. Here is the first of my attempts: The cylinders I denote $C_0 = S^1 \times [0,1]$ and $C_1 = S^1 \times [0,1]$. Let $T \subseteq R^3$ be the surface $$x = \cos…
goobie
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Neighborhoods of half plane

Define $H^n = \{(x_1, \dots, x_n)\in \mathbb R^n : x_n \ge 0\}$, $\partial H^n = \{(x_1, \dots, x_{n-1},0) : x_i \in \mathbb R\}$. $\partial H^n$ is a manifold of dimension $n-1$: As a subspace of $H^n$ it is Hausdorff and second-countable. If $U…
tom b.
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What is the characteristic property of surjective submersions?

In Lee's 'Introduction to smooth manifolds' he states that given smooth manifolds $X,Y$ and a surjective submersion $f:X\to Y$, then $f$ is a smoothly final map, that is for any further smooth manifold $Z$, and any map $g:Y\to Z$, we have $g$ smooth…
Mozibur Ullah
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Is every connected compact manifold homeomorphic to a ball with parts of its surface identified?

Part of what's motivating this question is the idea of putting "coordinates" on a manifold. The connection is that: given a manifold, if it is homeomorphic to an $n$-ball with parts of its surface identified, then it would be homeomorphic to an…
alphacapture
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