Episode #125 of the Stack Overflow podcast is here. We talk Tilde Club and mechanical keyboards. Listen now

Questions tagged [orientation]

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

Filter by
Sorted by
Tagged with
0
votes
1answer
217 views

What is the induced orientation on a 1-manifold with boundary that is the image of closed interval under a smooth immersion?

My book is An Introduction to Manifolds by Loring W. Tu. Pictured below is the last example from Section 22, Manifolds with Boundary. I have been trying to wrap my head around this for about 2 hours (...
1
vote
1answer
70 views

Is a smooth immersion $c: [a,b] \to M$ injective if its image is a 1-manifold with non-empty boundary?

My book is An Introduction to Manifolds by Loring W. Tu. Pictured below is the last example from Section 22, Manifolds with Boundary. In this question, it is confirmed that the example is an error. ...
1
vote
0answers
57 views

What is the induced orientation on a 1-manifold with boundary that is the image of closed interval under these maps?

My book is An Introduction to Manifolds by Loring W. Tu. Pictured below is the last example from Section 22, Manifolds with Boundary. In this question, Prof Jack Lee says that the example is ...
6
votes
2answers
707 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?
3
votes
1answer
222 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=\...
6
votes
1answer
110 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^...
1
vote
2answers
684 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
197 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 ...
31
votes
2answers
545 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-...
6
votes
2answers
2k views

Understanding the orientable double cover

Definition: if $M$ is a smooth manifold, define the orientable double cover of $M$ by: $$\tilde{M}:=\{(p, o)\mid p\in M, o\in\{\text{orientations on }T_pM\}\}$$ together with the function $\...
9
votes
2answers
379 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
2answers
347 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
votes
0answers
389 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 ...
8
votes
2answers
636 views

Partition of unity and volume form on a manifold

A smooth manifold $M$ is orientable iff there exists a nowhere-vanishing top form (i.e. volume form). In a coordinate chart $U\subset M$ we can find a volume form over $U$ that corresponds to the ...
7
votes
1answer
971 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
votes
2answers
330 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 ...
0
votes
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 ...
3
votes
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 $1$. ...
2
votes
2answers
140 views

Euler angles, Quaternion and mobile device rotation

I've written a JS SDK that listens to mobile device rotation, providing $3$ inputs: $\alpha$ : An angle can range between $0$ and $360$ degrees $\beta$ : An Angle between $-180$ and $180$ degrees $\...
2
votes
0answers
107 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 ...
1
vote
2answers
586 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?
0
votes
1answer
31 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 ...
0
votes
1answer
1k views

How to rotate an orientation (Euler angles)

If I have an orientation defined by Euler angles and I want to simulate a rotation of the coordinate system about the origin (doesn't matter to me how the rotation is specified), how would I get the ...
0
votes
1answer
707 views

How do I compare the (quaternion) orientation of two objects and specify different tolerances along each local axis?

I have two objects in 3D space with orientations represented by quaternions (but I can also convert to Euler angles if necessary) and I'd like to find out if they are oriented in more or less the same ...
5
votes
0answers
389 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
1answer
184 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 ...
1
vote
2answers
127 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 ...