# Questions tagged [orientation]

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

19 questions
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### Non-orientable 1-dimensional (non-hausdorff) manifold

Is there any nice example of a 1-dimensional non-hausdorff manifold that is not oriented? I have tried the line with two origins, but maybe something more exotic is needed?
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### Stokes' theorem: Induced orientation on the boundary of a manifold

The Question Let $K = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \geq 1\}$, where $K$ is oriented via the canonical volume form on $\mathbb{R}^3$: $dx \wedge dy \wedge dz$. Let $\mathbb{S}^2$ be ...
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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 ...
### The identity map from $\mathbb{\bar B}^3$(as a subset of $\mathbb{R}^3)$ into $\mathbb{\bar B}^3$(as a smooth manifold with boundary) is not smooth?
Let $U$ be the open rectangle $(0, \pi) \times (0,2 \pi) \subset \mathbb{R}^2$ and let $X : U \rightarrow \mathbb{R}^3$ be the following map: X(\varphi , \theta)=(\sin \varphi \cos \theta , \sin ...