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Questions tagged [manifolds]

A manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.

A manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.

534 questions
26
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1 answer

Classification of 1-dimensional manifolds (not second-countable)

It is easy to see that every connected $1$-dimensional second-countable manifold (that is, what is often called just a manifold) is either homeomorphic to $\mathbb{R}$ or to $S^1$. Now let's drop the secound-countable-condition. How do you prove…
Martin Brandenburg
  • 61,443
11
votes
2 answers

Cardinality of connected manifolds

Consider the assertion: Every connected, but not necessarily paracompact, n-manifold is of cardinality $2^{\aleph_0}$ (at least assuming the axiom of choice). For n=1 this may be proved via enumeration of the short list of examples. The essential…
Adam Epstein
  • 2,440
7
votes
0 answers

a "homological dimension" for embedding of manifolds

Let $A\to B$ be a surjective map of commutative $k$-algebras, and suppose $C\to B$ is a free resolution of $B$ as an $A$-algebra, meaning that $C$ is a free non-negatively graded commutative $A$-algebra with a differential decreasing degree by 1,…
Dima Roytenberg
  • 378
5
votes
5 answers

Background to learn about manifolds

Greetings As a necessity to go forward with physics, I find myself in the need to learn about manifolds. Being an engineering student, I don't have the chance to study topology in all its glory. So, can any one point me to the right direction, i…
Toussaint
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5
votes
1 answer

transformation properties of divergence (of a vector field)

Hi, If I have a divergence free vector field defined on a smooth manifold, and I apply some diffeomorphism, what can I say about what happens to the vector field? The example I am using is of an open or closed (pick one) disk embedded in 3…
Ben Sprott
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5
votes
1 answer

Spins as tensor fields

I have often come across this implicit translation of the classical field of a particle of a given spin into a specific tensor field. But I could not locate any literature from which I could learn this. In a paper of Avrimidi I found this…
Anirbit
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5
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A question about something like "shelling" in a PL manifold

If $P$ and $Q$ are compact codimension zero submanifolds of a PL manifold, say that they meet nicely if $P\cap Q$ is a codimension zero submanifold of both $\partial P$ and $\partial Q$. In particular then $P\cup Q$ is a codimension zero…
Tom Goodwillie
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2
votes
1 answer

Decomposition of straight line between points on a manifold

In an article by Lubich, I came across a decomposition for points on the straight line between two points lying in an embedded submanifold $M$ of $R^{n}$. To be precise, it is proposed that for $X$, $\tilde X \in M$ and $\tau$ small enough, any…
Don Toddo
  • 23
1
vote
1 answer

Find a circle that intersects the image of $[0,1]$ in a manifold $\mathcal{M}$ at only 1 point

Let $\gamma : [0,1] \rightarrow \mathcal{M}$ be a continuous map so that $[0,1]$ is homeomorphic to $\gamma([0,1])$, where $\mathcal{M}$ is a manifold (Hausdorff, second countable, and locally Euclidean). Using a chart containing $\gamma(0)$, I…
Rahul Sarkar
  • 359
0
votes
0 answers

Locally flat submanifold

Recently I found the next definition: Let $M^n$ be an $n-$dimensional topological manifold. Then $N^k\subseteq M^n$ is a locally flat submanifold if for every $x\in N$ there exists an open set $U$ in $M$ such that the pair $(U,U\cap N)$ is…
Antonio
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0
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1 answer

What is the topology of the structure group of a fiber bundle?

I dont know how can we topologize the structure group G of a fiber bundle P:E\rightarrow B by transition functions \psi_i^j (Osbern) Do you know an easy and fundamental book on fiber and vector bundles? Thanks a lot
Hamid
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-2
votes
1 answer

A diffeomorphism between complex manifolds which is not a holomorphic map

Can someone give an example or a reference on this?
YuYang
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