Questions tagged [orientation]
For question regarding the notion of orientation both in topology and in global analysis.
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Jacobian of converting Euler angles to rotation vector or rotation matrix
please consider this paper: A Primer on the Differential Calculus of 3D Orientations - Bloesch 2016. Equations 27, 29 and 30, for example, give nice results about differentiating the rotation of a ...
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What is the correct term to describe direction of travel on a curve that intersects itself exactly once? [closed]
In a non-self-intersecting curve such as a circle, we can describe direction of travel as clockwise (CW) or counter-clockwise (CCW).
In a self-intersecting curve such as a figure-eight, clockwise is ...
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Regarding the winding number
My main question is about part B, but I would also be grateful if you can tell me what you think about part A.
Define a smooth vector field
$X$ on $S^1$ as follows:
$X(x,y)=(-y,x)$. For a smooth map $...
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Does a non-trivial orientable vector bundle with rank $n$ have $n$ independent sections?
Why I know
A bundle of rank $n$ is trivial iff it has $n$ linearly independent sections. We can write $B\times \mathbb R^n \rightarrow E(b,t_1,...,t_n)=\sum_i t_is_i(b)$ (where $t_i\in \mathbb R$)
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Covering transformation and orientation ( in Synge Theorem )
Case 1: $M_1$ is a compact orientable manifold. And
$$
\pi_1:\tilde M_1\rightarrow M_1
$$
is the universal covering of $M_1$. Introduce on $\tilde M_1$ the orientation and metric such that $\pi_1$ ...
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$S^2$ is an orientable manifold
This question was left as an exercise in my class of orientable manifolds and I am having a hard time solving this.
Question: (a) Prove that $S^2$ is an orientable manifold. (b) Let M and N be two ...
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Every not-orientable $n$-manifold embeds in an orientable $(n+1)$-manifold
I have to prove that every not orientable smooth $n$-manifold $M$ can be embdedded in an orientable smooth $(n+1)$-manifold. My idea is to use the embedding theorem of Whitney but it just says that ...
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Evaluate the surface integral over a torus
Let $F(x) = \frac{x}{||x^3||}$ and let $B$ be the 3D region formed when the circle $(z - 2)^2+y^2 = 1$, in the yz plane, is rotated about the y-axis. Evaluate $\int_{dB}F \cdot dS$.
This shape $dB$ ...
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Tangential Circles connecting Oriented Points
I have two arbitrary points (A and B) in a plane, and I'm working in cartesian coordinates.
Each of these points has an associated arbitrary unit direction vector - so from point A, for example, I ...
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Question about the definition of orientation in Vector Calculus and Differential Forms: A Unified Approach" by Hubbard.
I am trying to understand the definition of orientation as it is done in the book "Vector Calculus and Differential Forms: A Unified Approach" by Hubbard. What I really like about it is that ...
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Orientation of the Space of Self-Dual Forms
Let $X$ be a closed, orientable, smooth 4-manifold. Let us give an orientation to $X$ by setting $e^1, e^2, e^3, e^4$ to be an oriented local orthonormal basis for its cotangent bundle.
Does this ...
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Induced orientation on the boundary of a $1$-dimensional manifold
Consider a $1$-dimensional smooth compact manifold with boundary $M$. Since $M$ is $1$-dimensional, then $\partial M$ is a $0$-dimensional manifold and it will have an induced orientation. I want to ...
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Theorem about deformations of positively oriented vectors in $\mathbb{R}^{n}$
I'm reaching out to you because of theorem 1.17 in the textbook "Multivariable calculus with applications". The theorem states the following:
Theorem
Every positively oriented ordered list ...
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Definition of orientation on manifolds and global continuous frame
In Tu and Lee books on smooth manifolds the definition of orientation on a manifolds $M$ is given by
We first assign a pointwise orientation : for every $p \in M$ we choose a class of oriented basis ...
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What is meant by reversing the orientation of an oriented manifold?
Trying to understand the following statement, I hope my stupid question won't bother you too much. Thank you.
Proposition. Let $M$ be an oriented smooth $n$-manifold and let $\omega$ be a compactly ...
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Finding a subgraph satisfying degree constraints in a directed graph
We are given a directed graph $D=(V,A)$ and two values $i(v)$ and $o(v)$ for each vertex. Is it NP-hard to find an induced subgraph of $D$ such that the in degrees are at most $i(v)$ and the out ...
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Can a diffeomorphism between connected smooth manifolds be both orientation perserving and reversing?
Can a diffeomorphism between connected smooth manifolds be both orientation perserving and reversing? i.e. preserving orientation at one point, but revsersing at another point?
I'm reading a book on ...
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Representing reflection matrix as rotation matrix
I have a reflection matrix that represents the orientation of a particular domain. The determinant of this matrix is (-1). Is it possible to rewrite this reflection matrix in the form of a rotation ...
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Orientability and unit tangent bundle of surfaces
What can we say about the orientability of the unit tangent bundle $ UTM $ of $ M $?
The unit tangent bundle of the sphere $ S^2 $ is $ \mathbb{R}P^3 $ see A question on the unit tangent bundle of the ...
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Orientable vs Oriented connected sum
I am confused about defining the connected sum of two $\color{red}{oriented}$ $n$-dimensional connected smooth manifolds without boundary, where $n\geq 2$. Could anyone help me to clear my confusion? ...
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Is the existence of a continuous top form sufficient for orientability of a smooth manifold?
I'm currently working through Tu's Introduction to Manifolds. He first defines orientability in terms of the existence of orientations on the tangent spaces which can be locally represented by a ...
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Prove that $\mathbb R\mathrm{P}^n$ is orientable if and only if $n$ is odd
I am trying to prove that:
The real projective space $\mathbb R\mathrm{P}^n$ is orientable if and only if $n$ is odd.
For do so, consider first the antipode map $\sigma:\mathbb R^{n+1}\to \mathbb R^{...
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Explicit description of the homology of the n-sphere
I'm currently studying the homology of manifolds (so it's mainly orientation and duality questions) and I was wondering how to explicitly compute homology of spaces and particularly $n$-spheres.
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Orientation of left and right translations on a Lie Group
I read that if $G$ is a connected Lie Group, then for every $g \in G$, the left and right translations ($L_g$ and $R_g$, respectively), preserve orientation. The reason given is that $L_g$ and $R_g$ ...
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orientability of surface and odd/even no. of punctures [closed]
Suppose that T is a surface with Euler characteristic -4.
Is it orientable or non-orientable?
Does it have odd or even number of punctures (disks)?
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Quaternions to keep track of orientation: The effective quaternion does not reproduce the same results as multiple quaternions sequentially.
In short, I am trying to find the preferred orientation of a complex non-symmetric object on a surface using a Monte Carlo method. During each cycle, the object is moved with a random (limited) ...
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Fluid dynamics - Orientation of a mathematical plane surface
The following is a problem in "Transport Phenomena" by Bird et al :
"A mathematical plane surface of area $S$ has an orientation given by a unit normal vector $\bar{n}$, pointing ...
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Is this a valid definition of an orientable manifold?
That is, an n-dimensional manifold M is said to be orientable iff the tangent spaces of its points are consistently oriented(i.e. for any 2 distinct points p and q, the matrix $B_{pq}$ which ...
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Fundmental class of a homology manifold
Suppose $X$ is a compact, connected $n$-dimensional homology ($\Bbb Z$-)manifold (https://en.wikipedia.org/wiki/Homology_manifold). Since orientability is defined using only homology (for example, in ...
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What is the induced orientation on a product of vector spaces in singular cohomology?
In singular cohomology, we understand an orientation of $\Bbb{R}^d$ as a choice of generator for the relative cohomology group $H^d(\Bbb{R}^d,\Bbb{R}^d\setminus \{0\})\cong \Bbb{Z}$.
My question is ...
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Intuition for orientation of a simplex (in 3 dimensions)
In trying to begin to learn basic homological algebra, i am confronted with orientation of simplices. The definition seems unmotivated and unintuitive: for $n$-simplices with $n \in \{-1,0,1,2\}$, it ...
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Transitive action on non orientable $ M $ lifts to orientable double cover
Suppose that $ M $ is non orientable with transitive action by a Lie group $ G $. Does that imply that some Lie group $ G' $ acts transitively on the orientable double cover $M'$?
This is true for ...
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Oriented surface, oriented curve, and vector field
I have an oriented surface S and an oriented curve C. The surface and the curve intersect in points A and B. In A, the orientation of the curve is the same of the surface, in B they are in opposite ...
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Can I compute contour orientation without using polygon area sign?
Most of the times, I determine contour orientation generating 2D points and computing the closed polygon area. Depending on the area value sign I can understand if the contour is oriented clockwise or ...
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Intuitive understanding of oriented volume and trivectors
I get that the way a vector's arrowhead points corresponds to its orientation for a given direction (line). We can also understand vectors within $V$ as isomorphic to a set of endomorphic translations....
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Orientability of surface r(u,v)=(ucos(v),usin(v),v)
Say one wanted to take the surface integral $\iint_S F\cdot dS$ of the vector field $F(x,y,z)=\langle z,y,x \rangle$ over the surface $S$ parametrized by $r(u,v)=\langle u\cos(v),u\sin(v),v\rangle$, $...
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Degree of a map between different dimensional manifolds
Let $f: M^m \longrightarrow N^n$ with $m=n$. The degree of $f$ can be defined as $\deg(f) \in \mathbb{Z}$ such that $f_*\big([M]\big) = \deg(f)\cdot [N]$, where $[M]$ and $[N]$ denote the fundamental ...
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How to express a rotation to describe the change of orientation between two quaternions?
Please let me describe the following situation:
Having a quaternion q1 representing an orientation of an object in 3d space at time t=n and one other q2 at time t=n+1.
Is it possible to calculate a ...
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How to solve the inverse kinematics of a 2DOF system on a rotating platform?
I have a 2DOF (z,y axes) stabilization system that needs to maintain the orientation of the end-effector.
Suppose the system is attached to a 3DOF rotating platform and has some fixed joints at the ...
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How to find the rotation matrix (with no x rotation) between two rotation matrices?
I need to find the rotation matrix (with no $x$ rotation) between two rotation matrices.
Given a starting rotation matrix $\textbf{R}_a$ and a setpoint $\textbf{R}_{SP}$. I need to find the rotation ...
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Checking a quotient space homeomorphic to torus
Let $\Bbb S^1:=\{z\in \Bbb C:|z|=1\}$, $\Bbb T:=\Bbb S^1\times \Bbb S^1$, $\Bbb A:=\{z\in \Bbb C:1\leq |z|\leq 2\}$, and $\Bbb C^*:=\Bbb C\backslash \{0\}$.
Suppose we are given the following data:
$(...
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Exercise 5-15 from Spivak's Calculus on Manifolds
I came across the following question (Exercise 5-15) in Spivak's Calculus on Manifolds and am not sure how to solve it.
Let $M$ be an $(n-1)$-dimensional manifold in $\mathbb{R}^n$. Let $M(\epsilon)$ ...
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Regarding the Definition of an Orientable Foliation on a Manifold in terms of Transition Maps
Introduction to the Geometry of Foliations, Part $A$, Authors Gilbert Hector and Ulrich Hirsch, Page $15$.
Let $\mathcal{F}$ be a Foliation on a Manifold $M$ defined by the atlas $\{(U_i,\phi_i)\}$.
...
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Example of orientable surface $M\subset\mathbb{R^n}$ such that has no normal vector fields
There is the following well known result about manifolds in $\mathbb{R^n}$:
Theorem: Let $M\subset\mathbb{R^n}$ be a surface (manifold) of class $C^k$ $k\geq 1$ and dimension $m$. If there are $n-m$ ...
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Could there be a notion of orientation for domains of multi-dimensional integrals?
Other answers have explained that the reason why we take the absolute value of the Jacobian $|J|$ in multi-dimensional integrals but not in 1D integrals is because in 1D integrals a change of ...
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Fundamental class of $M\times N$
(Hatcher 3.B.4) Show that the cross product of fundamental classes for closed $R$-orientable manifolds $M$ and $N$ is a fundamental class for $M\times N$. Assume $\dim M = m,\dim N = n$.
Let $[M]$ ...
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Show that $M/G$ is orientable
(Hatcher Exercise 3.3.4) Given a covering space action of a group $G$ on an orientable manifold $M$ by orientation-preserving homeomorphisms, show that $M/G$ is also orientable.
I first assumed $\{\...
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How does a complex oriented cohomology theory induce a formal group law, grading issues.
For a complex oriented cohomology theory $E^*$, with complex orientation $t\in E^2(\mathbb{CP}^\infty)$, I am under the impression the following facts are true:
For formal reasons (the degeneration ...
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If $M\setminus\{p\}$ is orientable then $M$ is orientable.
I have a question on Hatcher exercise 3.3.2 which says that
Show that deleting a point from a manifold of dimension greater than $1$ does not affect orientability of the manifold.
There is a post ...
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How to prove that, in the positive orientation, the surface is always on the left?
Let $S$ be an oriented surface with unit normal vector $\mathbf{n}$ and parameterization
$$r(u,v)=(X(u,v),Y(u,v),Z(u,v)),\qquad (u,v)\in D\subset\mathbb R^2.$$
In the context of vector calculus, the ...
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