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Questions tagged [orientation]

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

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262 views

Show that at regular surface is orientable if it has a smooth normal vector field

Let $S\subset\mathbb{R}^3$ be a regular surface, and let $N:S\to S^2$ be smooth. I want to show that then $S$ is orientable. My definition of $S$ being regular is that it is a smooth surface such ...
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54 views

If $\phi$ is a orientation preserving linear automorphism, do we really need to choose the same orientation for the domain and the codomain?

In the book of Linear Algebra by Werner Greub, at pages $131-132$, it is given that and However, when $\phi$ is a linear automorphism, it says that $\Delta_F = \Delta_E$, and derives the following ...
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173 views

can't compute rotation from 3d rotation matrix just from eigenvalues and eigenvectors

Starting with a $3 \times 3$ rotation matrix $R$, I would like to know the axis and angle of rotation. It seems like a popular topic for questions on this forum, but I can't quite find the answer to ...
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41 views

No closed surface diffeomorphic to the Moebius trip without boundaries

Consider the Moebius strip $M$ as a quotient space of $\mathbb{R}\times [0,1]$ with opposite lines glued together with reverse orientation. I would like to prove that a closed submanifold $N$ in $\...
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104 views

Hint for proving that there are exactly two equivalence classes under the relation being consistently orientable

Let $V$ be a real vector space of dimension $n \geq 1$. We say that two ordered bases $(v_1,\dots,v_n)$ and $(\widehat{v}_1,\dots,\widehat{v}_n)$ are consistently oriented if $\det B > 0$ ...
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493 views

Calculate orientation quaternion given two axes of a coordinate system

I know how to use quaternions to rotate my coordinate system about an axis and how to chain those together. Since this is not too complicated I thought it would also be easy to go the other way around,...
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59 views

For all orientation-preserving homeo is there a homotopic orientation-preserving diffeo?

Basically, I want to know if the following is true: Given an orientantion-preserving homeomorphism $f:R\to S$ between Riemann surfaces $R$ and $S$, does exist an orientation-preserving ...
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161 views

orientation independent of triangulation

I have $2$-dimensional compact connected orientable manifolds... It is known that those can be triangulated in such a way that everything fits nicely: number of triangles is finite, each edge is edge ...
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109 views

Prove there exists a smooth unit normal at the boundary of the following manifold

Let $M$ be a compact subset of $\mathbb{R}^3$ with smooth boundary $S=\partial M$. Consider M with the standard orientation $\mu=\mu_{0}$ from $\mathbb{R}^3$ and $S$ with the boundary orientation $\...
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637 views

Does stereographic projection preserve or reverse orientation?

Let $S^n\subset\mathbb{R}^{n+1}$ denote the standard unit sphere with normal bundle $\nu$, let $N=(0,\dots,0,1)$ and $S=(0,\dots,0,-1)$. Then there are two sterographic projections $$\sigma_+\colon S^...
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satellites attitude determination TRIAD - how are orbital reference frame vectors constructed?

I posted this same question on space.stackexchange but never received any answer. So I am posting here hoping to get an answer as this is a quite mathematical topic. I am trying to fully understand ...
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30 views

understanding orientable manifolds

I'm reading Warner. "Foundations of Differentiable Manifolds and Lie Groups." p. 138. I don't get the statement in the definition of orientable manifolds. 4.1 Definitions $\;$ (the preface omitted) ...
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Finding and Comparing 2 Sensors Rotations, with same reference frame but different initial Orientation

Let's say we want to Compare two different Arm (Humerus) Rotations (series of quaternions) and we do not care about space translation but only for rotation. To measure each rotation we use the same ...
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35 views

The orientation induced on the boundary of a manifold.

I just learned about the notion of orientability of a manifold which is difficult and abstract for me. If we consider all basis of a vector space, the matrix that transforms one basis in another basis ...
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40 views

Prove that an integration is left invariant.

$G$: Lie group of dimension $n$. $\tilde{\Omega}$: Orientation on $G$. $\Omega=\epsilon^1\wedge \epsilon^2\wedge \cdots \wedge \epsilon^n$ where $\epsilon^1\, \epsilon^2, \cdots ,\epsilon^n$ is ...
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36 views

Orientation of 4 + 1 lines in $\mathbb{R}^3$.

I'm working on a 3D algorithm that at some point establishes orientation of two lines - the same way one would do using the triple product. The way those lines are described, however, makes the ...
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97 views

How Euler angles change when we reverse direction of some axes

I looked at the other questions but couldn't find an answer for this particular question: I have measured Euler angles of an object in one coordinate system and I need to use these data in another ...
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1answer
33 views

Compatibility of orientation classes

Let $U$ be an oriented topological manifold of pure dimension $n$ and $K$ a compact of $U$. There is an orientation class $$or_{U,K} \in H_n(U, U \backslash K).$$ Let $V$ an open subset of $U$ such ...
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Orientation of the sum of displaced 2d gaussians

I'm interested in finding the orientation of the sum of 2d gaussians. If one gaussian is placed at the origin, and another displaced along the x axis, the sum of the two is going to have an ...
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42 views

Find rotation matrix for oriented robotic arm

I have a robotic arm centered at origin which I want to move from point A to point B. The robotic arm has an initial orientation matrix Rm and initial rotation matrix R. I also have a function that ...
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58 views

Criterion for being in a non-orientable 3 manifold?

I'm trying to wrap my head around the concept of orientability as an intrinsic property of a manifold. Assume I'm in some (3-dim) manifold for which I'd like to decide its orientability; what could I ...
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178 views

Proving that a complex manifold is orientable.

I am in the process of proving that a complex manifold is orientable. Consider the case $m=1$ so that in some chart, the usual coordinates of $p\in M$ are $(x,y)$. In some overlapping chart, let the ...
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proof given for spheres to be done for manifolds

All I am looking at the following theorem in the book of Bredon and my question is how does it work for manifolds instead of spheres? The proof is understood, but I am wondering if it is enough to ...
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71 views

Sphere eversion in $\mathbb R^4$

I know it's possible to perform a sphere eversion is $\mathbb R^3$, if we allow self-intersections. My question is: it's possible to perform a sphere ($S^2$) eversion in $\mathbb R^4$, without self-...
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Transition function and orientation-reversing patches on a nonorientable manifold

Let us consider gluing many patches to obtain a nonorientable manifold. If we have an intersection of 3 patches, the transition functions must be consistent on this intersection, i.e. there is some ...
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185 views

Möbius Strip is no orientable

This is an exercise from Do Carmo's Riemannian Geometry book. Let $G=\{Id,A\}$, $C= \{ (x, y, z) \in \mathbb{R}^3; x^2 + y^2 = 1, -1 < z < 1 \}$, where $A(p)=-p$. Define $\frac{C}{G}$ the ...
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152 views

(Non)-orientable surfaces and (non)-coorientable surfaces (and a little bit of physics)

I (think I) know the difference between orientable and non-orientable topological surfaces. I don't know the difference between co-orientable and non-coorientable surfaces. I must admit that I am not ...
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55 views

Finding the flux of the vector $ \vec F=z\hat k$ across the boundary of a sphere centered around the origin having positive orientation

I'm trying to find the flux of the vector $ \overrightarrow F=z\hat k$ across the boundary of a sphere $S$ centered around the origin with radius $\\a\\$ and having positive orientation. The answer is ...
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60 views

Is $M:= \{(x,y,z) \in \mathbb R^2: x^2 +y^2 = 1, x+y+z=0 \}$ an oriented smooth manifold?

Define $M:= \{(x,y,z) \in \mathbb R^2: x^2 +y^2 = 1, x+y+z=0 \}$ and define $\Phi_1, \Phi_2:(0,1)\to M$ by $\Phi_1(t):= (\cos(2\pi t),\sin(2\pi t),-\cos(2\pi t)-\sin(2\pi t)),$ $\Phi_2(t):=(\...
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154 views

Show that the boundary orientation agrees with the standard orientation of $\mathbb R^{k-1}$ if and only if k is even.

$H^{k}$ is oriented by the standard orientation of $\mathbb R^{k}$. Thus $\partial$$H^{k}$ acquires a boundary orientation. But $\partial$$H^{k}$ may be identified with $\mathbb R^{k-1}$. Show that ...
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135 views

How to Define Orientation of a Topological Manifold via Sheaves

I have just started reading orientation of a topological manifold from Hacther's Algebraic Topology. It was hinted by one of my professors that orientation of a manifold can be looked at from the ...
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37 views

Interpretations of orientability on higher order homology groups

A surface $S$ is orientable iff its first homology group $H_1(S)$ is a free abelian group $F$, or it is non-orientable iff $H_1(S)$ has the form $F + \mathbb{Z}_2$ (so says wiki, "orientability and ...
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187 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 ...
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70 views

Exact top forms are not volume elements

Let $M$ be a compact, orientable, smooth manifold with empty border. Prove that a volume element $\omega$ of $M$ cannot be an exact form (i.e., there is no $\eta\in\Omega^{n-1}(M)$ such that $\omega=\...
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121 views

Exercise about a 1-form on a manifold

I'm struggling a little bit thinking about this apparenlty innocent exercise. I'll provide an incomplete solution. Exercise Let $M$ be an $n$-dim manifold and let $S\subset M$ be an embedded $(n-1)$-...
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27 views

How to determine if an object in space is pointing at (oriented toward) another object?

QUESTION: You know the position of two objects in space (one also has an orientation). How do you determine when the object is pointing/oriented at the other object? Hopefully this question makes ...
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148 views

Normal vector field for an immersion

I know that a hypersurface $M$ of a riemannian manifold $N$ is orientable iff there exists a globally defined unit normal vector field $\eta : M \to TM^{\perp} \subset TN|_M$. Does the same hold for ...
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90 views

If a surface has a differentiable Gauss map, then it has an orientation?

If a surface $S\subset R^3$ has a differentiable Gauss map $N:S\rightarrow S^2$, then $S$ has an orientation? How can I prove this statement? (Here, orientation is defined by a choice of equivalence ...
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103 views

How can I calculate the integral $\int_M F^* \omega$?

I got stuck in the following problem. Let $M$ be the manifold defined by the equation $x^2+y^2+z^4=1$ and $F: M \to S^2$ defined as $F(x,y,z)=(x,y,z^2)$. I have to calculate the integral $\int_M F^* \...
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39 views

Compatibility of two notions of orientability for bundles

I have come across two notions of orientability : Notion 1 : A smooth manifold $M$ of dimension $n$ is said to be orientable iff $\exists$ a nowhere vanishing smooth $n-$form. The other notion in ...
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48 views

Does Translation Induce the Same Generator as The Ball?

$\newcommand{\R}{\mathbf R}$ Let $x$ and $y$ be distinct points in $\R^n$ and $B$ be an open ball containing both $x$ and $y$. Let $i:(\R^n, \R^n-B)\to (\R^n, \R^n-x)$ and $j:(\R^n, \R^n- B)\to (\R^n, ...
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Poincaré Duality as a $G$-morphism

In Brown's Cohomology of Groups book, at page 211 there is the following statement (Prop. 8.2): Let $Y$ be a compact $d$-dimensional $K(G,1)$-manifold (possibly with boundary). Let $X$ be its ...
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621 views

Convert from Rotation Matrix to Fixed-Axis (Extrinsic) zxz Euler Angles

How can an arbitrary rotation matrix $R = \left(\begin{matrix} R_{11} & R_{12} & R_{13} \\ R_{21} & R_{22} & R_{23} \\ R_{31} & R_{32} & R_{33}\end{matrix}\right)$ be ...
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87 views

The trivial cobordism and orientations

I have just come across the definition of a cobordism between two closed oriented $n$-dimensional manifolds $M,M'$ as an oriented $(n+1)$-dimensional manifold $W$ with boundary such that $\partial W =...
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107 views

Surjective map from orientable covering space to orientation cover of base space.

Let $p \colon M \to N$ be a covering space, and let $M,N$ be manifolds. Assume now that $M$ is orientable and $N$ is not orientable. I'm asked to find a covering map $q \colon M \to N$ s.t. $p= \pi ...
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602 views

How to determine the outward normal vector of a face of a hexahedron if the orientation (CW or CCW) of vertices of the face is unknown

Consider a single quadrilaterally-faced hexahedron. If given the co-ordinates of the vertices, $\mathbf{v}_i$, of a face in counter-clockwise orientation, I can compute the corresponding unit outward ...
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261 views

Orientation twisted on the $2$-sphere

If one is asked to verify that the canonical orientation on the $2$-sphere $S^2$ is twisted by the antipodal map, is it right to say that this is so simply because the Jacobian matrix of the map $A\...
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Smooth isotopy preserves orientation

Let $N$ a $n$-dimensional connected manifold and let $h: N \rightarrow N$ a diffeomorphism such that $h$ is smoothly isotopic to the identity map $\text{id}_N : N \rightarrow N$. It's clair that the ...
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56 views

Can it be Shown That an Angle $+ 2\pi$ = the Same Angle?

This might be some kind of foundational assertion. Given any angle $\psi$ and $x\in\Bbb Z^+$ I'd like to see proof that: $$\psi = \psi + 2x\pi$$ It's hard cause numerically this isn't true, but in ...
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58 views

Why do left and right switch when direction is reversed? [closed]

If I make a left turn during a trip, it becomes a right turn on my way back. Why is this?