# Questions tagged [manifolds]

For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

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### Manifolds and covering maps [duplicate]

I am studying topological manifolds; I know that if $M$ is an $m$-topological manifold and $p:M\rightarrow N$ is a covering map, then $N$ is an $m$-manifold. I know how to prove the existence of local ...
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### M is composed of line segments connecting ellipse to $(0,0,0)$ Calculate integral $\int_M \sqrt{x + 3z}\ d \lambda_2$ over those. Almost done.

I found such an exercise among my set of exercises preparing for exams and I have no idea how to solve that. Every point of ellipse $\{ (x,y,z) \in \mathbb{R}^3 : x^2 + y^2 = 1, x + z = 1 \}$ is ...
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### Confusion on Notations of Partial Derivatives on Manifolds

I'm confused with the notations of partial derivative on manifolds in Tu's An Introduction to Manifolds.. Just to make clear the notations I'm using, what I've known and which part I'm confusing, ...
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### Find a topological manifold that is not Hausdorff and not locally compact

Define a topological manifold as a space locally homeomorphic to $\mathbb{R}^n$. Find a topological manifold that is not Hausdorff and not locally compact. (Hint:Consider $\mathbb{R}$ with "extra ...
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### Each elements in Locally finite precompact open cover of a topological manifold has at most finite intersection with others

The problem is from John M Lee's Introduction to Smooth Manifold:problem 1.4.(see below image) I have checked similar post regarding this problem from elsewhere of this site. However, I still find it ...
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### Can the twice-punctured plane be given a homogeneous metric?

A homogeneous metric on a space $X$ is one for which the isometry group acts transitively on its points (for all $x,y\in X$, there is an isometry $\phi$ of $X$ such that $\phi(x)=y$). If we repeatedly ...
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### Open subset of paracompact manifold is paracompact?

I assume a smooth manifold is paracompact (including Hausdorff) instead of Hausdorff and second countable. We can show that a connected topological manifold is paracompact iff it's second countable. ...
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### What is the theory of the measure of a line in a plane

I have been taught how to measure the length of a line in a plane, in typical situations encountered in physics; for instance, by integrating $dl = \sqrt{dx^2+dy^2}$ along the line, parametrized in ...
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### What's the continuous $\mathbb C^0$ version for immersion? I'm trying to see if topological embeddings are 'topological immersions.'

A 'smooth embedding' $f: M \to N$ between smooth ($m$,$n$)-manifolds ($M$,$N$) is a smooth map that is both a smooth immersion and a topological embedding, which is simply defined as that the range-...
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### Can locally Euclidean Hausdorff 2nd countable topological space always be made into topological manifolds (as in continuous or $C^0$)?

For any locally Euclidean Hausdorff 2nd countable topological space $M$, can $M$ always be made into a topological manifold (not necessarily of some uniform dimension $n$ ... but if ever I assume ...
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### Do smooth manifolds care about the continuous atlas?

A smooth manifold $M$ can be defined as a pair (topological manifold $X$, smooth, as in $C^{\infty}$, atlas $\mathscr M$), where a topological manifold is defined as a locally Euclidean Hausdorff 2nd ...
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### Using Implicit Function Theorem to Find a Coordinate System near a point of a level set map

Let $F: \mathbb{R}^n \to \mathbb{R}^m$ be a smooth function, $c \in \mathbb{R}^m$ be a regular value and $S=F^{-1}(c)$ be the embedded submanifold given by the regular level set. It is well-known that ...
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### Show that the given stereographic projection is $\sigma(x)=u$

Let $N$ be the North pole $U_N=(0,\dots, 0, 1)$ and the South pole $U_S=(0,\dots, 0, -1)$, where both are $\in \mathbb{S}^n\subseteq \mathbb{R}^{n+1}$. Define the stereographic projection \$\sigma:\...
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