Mathematics Stack Exchange
Mathematics Stack Exchange

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.

8723 questions
2
votes
1 answer

Proof of triangulation of $1$-manifold in John Lee's book

In "Introduction to Topological Manifolds" by John M. Lee, the following in stated in the proof of Theorem 5.10 pp. 102. Note that $\text{Int } e\cap \text{Int } e'$ is open in $\text{Int } e$. On the other hand, $e'$ is a compact subset of the…
Jeremiah Goertz
  • 141
2
votes
0 answers

How can two charts in a manifold be mutually contradictory?

In page 7. of Fecko's book on Differential Geometry and Lie Groups, he says, In an effort to map a bigger part of a country, an atlas (a collection of maps) has proved to be helpful. A good atlas should be consistent at all overlaps: if some part…
Ron
  • 635
2
votes
0 answers

What are the (non-piecewise) linear manifolds?

For all non-negative integers $n$, $B_n$ is defined to be $\: \big\{\mathbf{v} \in \mathbf{R}^n : ||\mathbf{v}||<1\big\} \:$. For what manifolds does there exist an atlas of charts $\: c : U\to B_n \:$ such that the transition maps are all…
user57159
2
votes
1 answer

Differentiable manifolds: Definition of charts

I was reading "Introduction to differentiable manifolds" from Serge Lang, but I got immediately stuck. He defined his atlases as follows: Let X be a Hausdorff topological space. An atlas of class $C^P (p\geq 0)$ on X is a collection of pairs $(U_i;…
Dylan_VM
  • 579
2
votes
0 answers

Isomorphism between $H^{1}(S)$ and $Hom(\pi_{1}(S,s_{0}),\Bbb{R})$

Suppose $S$ is a connected two dimensional manifold(here we assume a manifold to be of countable basis),$s_{0}$ a base point of $S$. I was asked to show that the following map is an isomorphism: $$H^{1}(S)\to…
C Weid
  • 660
2
votes
2 answers

Equivalent definition of "immersion"

Definition: Let $M^m$ and $N^n$ be differentiable manifolds. A differentiable mapping $\varphi : M \to N$ is said to be an immersion if $d\varphi_p : T_p M \to T_{\varphi(p)}N$ is injective for all $p \in M$. Example: The curve $\alpha : \mathbb…
New day rising
  • 586
2
votes
1 answer

Suppose $M$ is Hausdorff and locally Euclidean. Show $M$ is second countable iff it is paracompact and has countably many connected components.

Struggling with paracompact & countably many components $\implies$ second-countable. We just need to show each component is second-countable. Choose a component $M_i$. $M_i$ is closed so it is paracompact. We can cover $M_i$ with precompact…
trystero
  • 701
2
votes
1 answer

Show that graph $ \Gamma(f) = \{x,f(x) : x \in \mathbb{R^n}\}$ of $f$ is a regular submanifold

Let $f : \mathbb{R^n} \rightarrow \mathbb{R^m}$ be a smooth map. Then show that the graph $\Gamma(f) = \{ (x,f(x) ) : x \in \mathbb{R^n} \}$ of $f$ is a regular submanifold of $ \mathbb{R^n} \times \mathbb{R^m}$ of dimension $n$. I need to use the…
user383517
  • 241
2
votes
0 answers

Why $M=\{(x, |x|), x\in\mathbb{R}\}$ is not an embedded submanifold?

How can I prove that $M=\{(x, |x|), x\in\mathbb{R}\}$ is not an embedded smooth($C^\infty$) submanifold of $\mathbb{R}^2$. I tried to say there is any ($C^{\infty}$) immersion from $\mathbb{R}$ into $\mathbb{R}^2$ such that its image is $M$, but I…
bigli
  • 29
2
votes
1 answer

how to prove that $C^{k}$ map does not depend on choice of the charts

I was reading an article about Manifolds.They have defined a $C^{k} $ function in the following way : Let $M$ and $N$ are two $C^{k}$ manifolds of dimensions $m$ and $n$ respectively.A continuous function $h : M \rightarrow N$ is said to be a…
Madhu
  • 1,695
2
votes
1 answer

Are there countably many closed manifolds in each dimension?

There is a single closed topological 1-manifold (up to, of course, homeomorphism): $S^1$. The classification of surfaces shows that there are countably many closed topological 2-manifolds. Classification of closed, orientable topological 3-manifolds…
Plutoro
  • 22,730
  • 3
  • 41
  • 71
2
votes
0 answers

Evaluate $\int_Mxdy\wedge dz$ where $M$ is the torus formed by the circle of radius $1$ in the $xz$ plane centered at $(2,0,)$ rotated around $y$ axis

Evaluate $\int_Mxdy\wedge dz$ where $M$ is the torus obtained by rotating the circle $(x-2)^2+z^2=1$ around the $y$ axis. I've parameterized $M$ using $\alpha:(0,2\pi)\times (0,2\pi)\rightarrow M$…
trystero
  • 701
2
votes
1 answer

degree of smooth maps from 2-sphere to 2-torus

Why any smooth map from the 2-sphere to the 2-torus has zero degree? Can we show that there is no surjective smooth map from 2-sphere to 2-torus?
Zeinab
  • 21
2
votes
0 answers

Does every smooth manifold admit a smooth $\Delta$-complex structure?

I know that every smooth manifold $M$ admits a triangulation. That means, there exists some simplicial complex $K$ homeomorphic to $M$. Does this mean that it also admits a smooth $\Delta$-complex structure? In Hatcher, this is defined as a…
David Roberts
  • 600
  • 3
  • 12
2
votes
1 answer

Is there restriction on the cardinailty of manifold?

I only know that the each point has some neighborhood with cardinality same as $\mathbb R$, but I have no idea about the cardinality of the whole manifold, must it have cardinality as $\mathbb R$? Please give some example to illustrate.
JSCB
  • 13,456
  • 15
  • 59
  • 123
1 2 3
…
23 24