Questions tagged [orientation]

For question regarding the notion of orientation both in topology and in global analysis.

295 questions
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On the unit sphere $S^2$, show the antipodal map $A:S^2\to S^2$ is orientation reversing using definitions.

I want to show that on the unit sphere $S^2$, the antipodal map $A:S^2\to S^2$ given by $(x,y,z) \mapsto (-x,-y,-z)$ is orientation reversing. I know that $S^2$ is a regular connected orientable ...
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Sard's theorem for orientation preserving diffeomorphism of the circle

thanks in advance for helping me. First I'll introduce some definitions: (1) Suppose that $f : \mathbb{S}^{1} \rightarrow \mathbb{S}^{1} = \mathbb{R} / \mathbb{Z}$ is an orientation preserving ...
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$M$ is orientable if and only if $M \setminus \{p\}$ is orientable.

Let $M$ a manifold of class $C^{\infty}$. Show that $M$ is orientable if and only if $M \setminus \{p\}$ is orientable. Comments: ($\Rightarrow$) Let $\omega: M \longrightarrow \Lambda^n(M)$ a ...
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Orientable cover of a non-orientable manifold factored through the orientation double cover.

While proving that orientable cover $M$ of a manifold non-orientable manifold $N$ factored through the orientation double cover, I got stuck in this following problem... If $p:N→M$ is a covering, N ...
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Definition of Integration of a Differential From in Lee's Introduction to Smooth Manifolds.

On pg. 402 of Lee's Introduction to Smooth Manifolds (Second Edition), the following is said to define the integral of a differential form on $\mathbf R^n$: Let $D$ be an open domain of integration (...
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Let $M$ is not orientable and $p \in M$, then $M - \{p\}$ is orientable.

Is it true or false? Let $M$ a manifold of class $C^{\infty}$ not orientable and $p \in M$, then $M - \{p\}$ is orientable. I believe this to be false, but I do not know a counterexample
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$R$ and $S$ homeomorphic Riemann surfaces $\implies$ $\exists h:R\to S$ orientation-preserving homeomorphism?

The question is quite that what is in the title: If $R$ and $S$ are homeomorphic Riemann surfaces, is it true that always exists a homeomorphism $h:R\to S$ which is orientation-preserving (at least ...
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Orientation and Coloring

Prove that the following two conclusion is equivalent for simple graph $G$. (i) The chromatic number $\chi (G) \leq k$; (ii) $G$ has an orientation where no directed path of length $k$ exists. ...