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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 decomposition of prime 3-manifold along incompressible tori help in classification of 3-manifolds

We know about geometrization conjecture. I am trying to understand how decomposition of prime manifold along compressible tori help classify 3-manifolds. Can we construct list of all prime manifolds this way ? Once we decomposed given prime…
mmm
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Showing that a manifold is locally compact

I am trying to show that every manifold (a Hausdorff and second countable topological space) is locally compact (has a precompact basis). Consider an open set of a manifold, $M$. I must show that for each $x\in U$ we can find an open set in $M$ such…
fosho
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Local extension of smooth function on regular manifold

Let $S\subset R^3$ be a 2d-regular manifold, and $g:S\rightarrow \mathbb R$ a smooth function. Show that for every $p\in S$ there is an open nbd $V\subset \mathbb R^3$ and smooth function $G:V\rightarrow R$ such that $G|_{S\cap V}=g|_{S\cap V}$. An…
S. R
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Orientation reversing involution on the sphere

This is the explanation from Jänich on why the antipodal map $\tau:S^n \rightarrow S^n$ reverses orientation iff $n$ is even: For every $x\in S^n$, the differential of the diffeomorphism $-\text{Id}:D^{n+1}\rightarrow D^{n+1}$ takes the outward…
Lotte
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Neighbourhood in the definition of manifolds

I hope my question is not too naive. Sorry in advance if it is. When exploring the literature on topological manifolds, I found basically two main classes of definitions regarding the local "similarity" to $\mathbb{R}^n$. In the first definition,…
PAM
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Manifolds and level sets

Let $M$ be the set of points $(x,y,z)$ in $\mathbb{R}^3$ such that $x^2+y^2+z^2=1$ and $x^2=yz^2$. The point $(0,-1,0)$ is removed. The question is: after removing a second point (to determine), why is this a manifold? I can argue that each of the…
Isa
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Proof that a (smooth) manifold locally "looks like" a graph

Let $M \subseteq \mathbb{R}^{n}$ be an embedded (smooth) submanifold of dimension $m$, with $c \in M$. Show that there is an open set $U \subseteq \mathbb{R}^{n}$ containing $c$, and a smooth map $f : U \cap \mathbb{R}^{m} \to \mathbb{R}^{n-m}$,…
Bilbottom
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Coordinate patch of an n-sphere?

Possible Duplicate: Analogue of spherical coordinates in $n$-dimensions If we take a 2-sphere of radius a, we can define $ f(z, t) = (\sqrt{a^2-z^2}\cos t,\sqrt{a^2-z^2}\sin t, z) $ it's a parametrization of the sphere, if $ 0 < t < 2\pi $ and $…
Ormi
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Green's identities

I have the following problem: Let M be a compact connected riemannian manifold with $\partial M=\emptyset$ and $f \in C^{\infty}(M)$ and $\Delta f\geq0$. Show that f is constant. To show that f ist constant you can also show that $grad f=0$. Could…
Tobi92sr
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Quotient of $R $ by $2πZ$

Let the additive group $2πZ$ act on $R $ on the right by $x · 2πn = x +2πn$, where $n$ is an integer. Show that the orbit space $R/2πZ$ is a smooth manifold.
Ciris
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Prove：any smooth map $f:M \rightarrow \mathbb{R}$ can't be one to one.

Let $M$ be a connected smooth manifold, $\dim M \ge 2$. Prove：any smooth map $f:M \rightarrow \mathbb{R}$ can't be one-to-one.
Edward Grammer
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The smoothness of an inclusion map

Let $M$ and $N$ be manifolds and let $q_0$ be a point in $N$. Prove that the inclusion map $i_{q_0} : M \to M×N : p \mapsto (p,q_0)$, is $C^\infty$.
Ciris
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Is the condition that two manifolds share a (n-1)-manifold necessary to obtain that the union is also a manifold?

I looked on the site but I did not find anything matching my problem. I am working in 4D with cubical complexes, also known as Khalimsky grids; I mean that I work with "cubical" manifolds. Do you know if it is possible that, starting from two…
Nicolas Boutry
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Methods to find charts of manifolds

Are there some standard techniques to find charts and atlas of a manifold? I'm looking for an easy way to find charts (can be trivial ones) so that I could easily find more examples to play with and test some properties of manifolds.
user42912
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Verify that a set is a submanifold

I'm stuck on the following exercise: "Verify that $M:=\{(x,y,z)\in\mathbb{R}^3:x^2+3y^2+2z^2=3, x+y+z=0\}$ is a submanifold. Also, say what its dimension is and compute the Jacobian matrix of the function." The definition of $C^k (k\geq 1)$…
lorenzo
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