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Questions tagged [orientation]

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

6
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2answers
673 views

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?
6
votes
1answer
106 views

Expressing invertible maps $\bigwedge^{d-1} V \to \bigwedge^{d-1} V$ as $\bigwedge^{d-1}A$ for some $A$

Let $V$ be a real $d$-dimensional vector space, let $\bigwedge^{d-1} V$ be its exterior power. Consider the following claim: Proposition: If $d$ is even, then every invertible linear map $\bigwedge^...
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vote
2answers
540 views

Interpolating Between 2 Angles

I'm trying to understand how this works, and mathematically I'm having difficulty. Given 2 angles between $(-2\pi, 2\pi)$: $\theta$ and $\phi$ I want to interpolate between them by the ratio: r. My ...
1
vote
0answers
181 views

Manifolds with boundaries and partitions of unity

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 ...
30
votes
2answers
516 views

Orientability of $m\times n$ matrices with rank $r$

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-...
9
votes
2answers
325 views

A topological group which is also a (not necessarily smooth) manifold is orientable

I am trying to show that a topological group which is also a (not necessarily smooth) manifold is automatically orientable. I know of a proof involving transition functions for smooth manifolds, in ...
7
votes
1answer
291 views

Fixed points in mapping from Möbius strip to disk [Explanation or reference needed]

One of the most elegant demonstrations in topology is the proof of the inscribed rectangle problem (a solved variant of the unsolved inscribed square problem) which states that for any plain, closed ...
10
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0answers
374 views

Fundamental class of the connected sum of two closed orientable manifolds

I need to find a representation of $ [M \mathbin\sharp N] \in H_n(M \mathbin\sharp N) $ in terms of the fundamental classes $[M]$ and $[N]$. My idea is that $$ [M \mathbin\sharp N] = [M]+[N]$$ which ...
7
votes
1answer
886 views

Proving that the quotient manifold is orientable if and only if the group action is orientation-preserving

I'm trying to solve the following exercise in Lee's book. Suppose M is a connected, oriented smooth manifold and Γ is a discrete group acting freely and properly on M. We say the action is ...
4
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2answers
257 views

How does one orient a simplicial complex?

I have a simplicial complex, built out of hyper-tetrahedra (5-cells) with the topology of $S_{4}$ and I would like to assign an ordering to it's vertices (some couple thousand), so that I can apply a ...
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0answers
103 views

Continuity of point wise orientation

Prove that $[(X_1,...,X_n)]$ on a manifold $M$ is continuous if and only if every point $p$ in $M$ has a coordinate neighborhood $(U,\phi) = (U,x^1,...,x^n)$ such that for all $ q \in U$, the ...
3
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1answer
93 views

Let $p=(5,0,-4)$ and $v \in T_{(5,0,-4)}M$. Compute $(F^{*}\omega)_p(v)$.

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 $1$. ...
1
vote
2answers
531 views

Map isotopic to identity is orientation preserving

Let $M$ be an $n$-dimensional orientable and compact smooth manifold and $f:M\to M$ be a smooth map isotopic to the identity map. Is it true that $f$ is orientation preserving?
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1answer
29 views

On terms “Orientation” & “Oriented” in different mathematical areas?

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 ...
4
votes
1answer
189 views

Outward-pointing vector field on Projective space

Lee Smooth Manifolds problem 8-4 says that for every manifold with boundary there exists a smooth vector field that is outward-pointing when restricted to the boundary. Now if our manifold is $M=\...
4
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0answers
362 views

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 ...
1
vote
2answers
117 views

$R$ 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 ...
1
vote
1answer
162 views

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 ...
0
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1answer
1k views

How to calculate rotation quaternion between two orientation quaternions?

I have some device (3D pointer) connected to my computer which returns it's position (in cartesian XYZ system) and orientation (in quaternions). I receive this values about 30 times/sec. Now I need ...