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 the unit sphere, considered as the boundary of $K$, with the orientation on $\partial K$ given by the induced orientation […]

The only time that I’ve heard the term “orientation-preserving map” was in Linear Algebra, but today I read the term orientation-preserving homeomorphism of the circle in the following context: If a homeomorphism $f$ of $S^1$ has a rational rotation number $r$ ($r=\lim_{n\to\infty} \frac{F^n(x)-x}{n}$ where $F$ is a “lift” of $f$ to $\mathbb{R}$) and if $f$ […]

The goal of this question is to help to deal with different meanings of the words such as “orientation“ and “oriented” in different mathematical areas. Are different oriented concepts somehow related to each other in different mathematical areas? Oriented matroids (Matroid Theory) Oriented graphs (Graph Theory) Orientation (Topology, Global Analysis) Other areas?

Let me show my work before presenting the problem itself. Let $M=\{(x,y,z) \in \mathbb{R}^3 : x+y=5, x+z=cos^2y\}$. We can easily see that $M$ is a submanifold of $\mathbb{R}^3$ of dimension $1$. We can also see that we can construct a global atlas for $M$, say $\{(M,\varphi)\}$, where $\varphi:M\rightarrow\mathbb{R}$ is given by $(x,y,z) \mapsto y$. If […]

I know that $$M_{m,n,r} = \{ A \in {\rm Mat}(m \times n,\Bbb R) \mid {\rm rank}(A)= r\}$$is a submanifold of $\Bbb R^{mn}$ of codimension $(m-r)(n-r)$. For example, we have that $M_{2, 3, 1}$ is non-orientable, while some others are, such as $M_{3,3,1}$ and $M_{3, 3,2}$. Is there a way to decide in general if $M_{m,n,r}$ […]

How do I 1 show that $M=[0,3]\subset \mathbb{R}$ is a manifold with boundary? 2 find a $C^2$ partition of unity for the open cover $M=[0,2)\cup(1,3]$? 3 show that $\omega=(x-2)dx$ is/is not an orientation on $M$? What I know: Let $M$ be a manifold in vector space $V$. Then it is covered by coordinate patches $f:A\rightarrow […]

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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