Tagged Questions

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Defining a Hyperbolic Metric in a General Surface S

I hope someone can help me or give a ref. I'm trying to understand the general way in which one defines a hyperbolic metric on a given surface $\Sigma$ ( and, if not too complicated, on a manifold of ...
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Proving that charts are related

Let $A$ be an atlas on the set $M$ and let $x: U \to x(U)$ and $y : V \to y(V )$ be bijections from subsets $U, V \subset M$ to open sets $x(U), y(V ) \subset \mathbb{R^n}$. Show that if the ...
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prove that this function is an immersion

How I can show that $F: \mathbb{R} \rightarrow \mathbb{R}^2$ defined by $F(t)=(\cos(t),\sin(t))$ is an immersion? In my definition $F$ is an immersion if $\forall p\in\mathbb{R}$, $dF_p$ is injective. ...
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prove that a function is an immersion

How I can show that $F \colon \mathbb{R} \to \mathbb{R}^2$ defined by $F(t)= (\cos (t), \sin(t))$ is an immersion? In my definition $F$ is an immersion if $\forall p$,$dF_p$ is injective. I have ...
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Product manifolds

I have a question on the product of two manifolds. I have $M, N$ two real manifolds (with a smooth differentiable structure), with $\partial M=0$. I have showed that $M\times N$ has a natural induced ...
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homeomorphism between maninifolds

Exist a local homeomorphism between the manifolds with boundary $[0,1) \times [0,1)$ and $\mathbb{R}^{2}_{+}$? I don't think that a local homeomorphism like this can exist..
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integration of differential forms on covering space

Let $M_1,M_2$ be $n$-dimensional oriented manifolds. Let $f: M_1\longrightarrow M_2$ be an orientation-preserving diffeomorphism. Then for any $\omega\in \Omega^n_c(M_2)$ we have(page 85 of {Madsen: ...
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What is the best (in terms of effectively building understanding) direction from which to approach manifolds?

Thedore Frankel's book The Geometry of Physics presents Manifolds right away in Chapter 1 in the following manner: Introduce the Euclidean space $\mathbb{R}^N$ only as "the most important manifold". ...
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Source for Differential Manifolds/Geometry Questions?

I'm looking for a good source for a large collection of Differential Manifolds/Geometry questions covering a subset of the following topics: inverse function theorem, local coordinates, induced ...