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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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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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Id $f,g$ orientation preserving cricle-diffeomorphisms, then $\rho(g^{-1}\circ f \circ g) = \rho(f)$.

Let $F, G : \mathbb{R} \rightarrow \mathbb{R}$ be a lift of $f$ and $G$ of $g$. That is, $ \pi \circ F = f \circ \pi$ with $\pi(x) = e^{2\pi i}$. We define $$\rho_{0}(F) = \lim_{n\rightarrow \infty}\...
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Smallest rotation angle between quaternions accounting for symmetry

I am trying to compute a similarity measure between the orientation of 3D objects accounting for symmetry invariance. I have a set of 3D objects, which are defined by a center of mass, and 3 unit ...
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29 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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Sard's theorem for orientation preserving diffeomorphism of the circle

thanks in advance for helping me. First I'll introduce some definitions: (1) Suppose that $f : \mathbb{S}^{1} \rightarrow \mathbb{S}^{1} = \mathbb{R} / \mathbb{Z}$ is an orientation preserving ...
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If $(e_1, \ldots, e_n)$ and $(f_1, \ldots, f_n)$ are co-oriented, then so are $(e_1, \ldots, e_n, u)$ and $(f_1, \ldots, f_n, u)$

Suppose that $S$ is a smooth $n$-submanifold of $M$ where $\dim(M) = n+1$. Suppose also that $S$ and $M$ are both Riemannian and oriented. Suppose that $(e_1, \ldots, e_n)$ and $(f_1, \ldots, f_n)$ ...
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Show that $\mathbb{S}^{n+m}$ is not homeomorphic to a product of orientable manifolds

I want to prove that the sphere $\mathbb{S}^{n+m}$ is not homeomorphic to the product of N and M, orientable manifolfs with $\textit{dim}\;N=n$ and $\textit{dim}\;M=m$. I know that I have to use the ...
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How/why does the contraction of standard volume form give the canonical form.

$M \subset \mathbb{R}^{N}$ is a (oriented) $n-1$ dimensional submanifold. Suppose $\nu \in T_{p}M^{\bot}$, of length one (a normal unit vector on $M$). How and why does the contraction $\nu_{\neg}(...
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Submanifold of codimension 1 orientable iff there exists unit normal vector field.

Suppose I have a submanifold $M \subset \mathbb{R}^{n}$, of dimension $n-1$. Where a unit normal vector field is a section $\nu$ of the normal bundle $ TM^{\bot} \to M$. So the fibers are all the ...
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Every submanifold is orientable (co-dimension 1)?

Suppose I have a submanifold $M \subset \mathbb{R}^{n}$, of dimension $n-1$. Apparently it's orientable if and only if there exists a unit normal vector field on $M$. Where a unit normal vector field ...
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If $TM$ is trivial, then $\Lambda^n(M)$ is also trivial and $M$ is orientable

Suppose that $M$ is a smooth $n-$manifold. Suppose $$ TM=\coprod_{p\in M}T_pM $$ is the tangent bundle of $M$. And let $$ \Lambda^n(M)=\coprod_{p\in M}\Lambda^n(T_pM) $$ where $\Lambda^n(T_pM)$ is ...
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26 views

What is the direction of a rotation?

We usually talk about rotations in the clockwise or counterclockwise direction. But if a rotation is just a function defined on the space, then it is all about points and their images, and there is no ...
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Prove that the following function is an orientation of a curve

Assume that $f^{(j)}:\mathbb{R}^3\rightarrow \mathbb{R}$ for $j=1,2$ are some $C^1$ functions such that $(\nabla f^{(1)}, \nabla f^{(2)})$ are independent over the curve $\gamma = \{ x \in \mathbb{R}^...
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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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Create Wall 3D math oriented away from camera

I have 2 Points which has x,y,z let's say from and to I am drawing wall between them using ARkit ios To draw wall I use static ...
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Visualisation of an orientable surface bounded by the Möbius curve

I'm learning multivariable calculus on MIT OpenCourseWare. When the teacher explained Stokes theorem he mentioned the Möbius strip. He showed it was non-orientable. Then he showed a somehow twisted ...
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1answer
50 views

Orientable manifold $M$ ,then $\partial M$ is orientable

Let $M$ a topological manifold of dimension $n$ with boundary $\partial M$. We define $M$ to be orientable if $M- \partial M$ is orientable. Here when I say orientable, I mean there is a locally ...
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preserves or reverse orientation of sphere surface

Let $\varphi: (0, \infty) \times (0, \pi) \times (0, 2 \pi) \to \mathbb{R}^3 \setminus \{(x,y,z) \in \mathbb{R}^3| y=0, x \geq 0 \}$ $$(p,\phi,\theta) \mapsto (p \sin \phi \cos \theta, p \sin \phi \...
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Find the angles between two solids

I have 2 solids (A and B) and I need to find the three angles between their x, y, and z axes. If I calculate the geometrical center of the two solids (Ax, Ay, Az and Bx, By, Bz), is it correct to ...
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60 views

Orientation in the proof of Stokes Theorem

I'm reading the proof of Stokes theorem at page 83 of "Godinho, Natàrio, An introduction to Riemannian geometry" and I can't understand a passage in it, probably because the definition of ...
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1answer
27 views

Prove there exists a outward unit normal field on the boundary this manifold

Let $M$ be a compact subset of $\mathbb{R}^3 $ with the standard orientation $\mu =[e_1,e_2,e_3] $ and let $S = \partial{M}$ is its smooth boundary with the induced orientation from $M$. Prove there ...
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Null homologous loop and orientable surface

I am reading Algebraic Topology: A First Course written by Greenberg and Harper. On page 67 of this book it is stated that Let $\gamma$ be a loop in $X$ regarded as a map $f:S^1\to X$. For $\chi[\...
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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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What does it mean for a (non-smooth) homeomorphism between oriented smooth manifolds to be orientation preserving?

The definitions I know of orientability of manifolds are in terms of tangent spaces. However, for example in this answer there is mention of orientation preserving homeomophisms (between orientable ...
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Fundamental group of $M$ has no subgroup of index $2\Rightarrow M$ is orientable

Let $M$ be a connected smooth manifold such that, for every $p\in M$, the fundamental group $\pi_1(M,p)$ has no subgroup of index $2$. Prove that $M$ is orientable. Here's what I know: there is a ...
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27 views

Simple stokes - why is this the correct orientation?

We want to calculate $\iint_{S}\text{curl}(\vec F)dS$ where $\vec{F}(x,y,z)=(y^2z, xz,x^2y^2)$ and $S$ is the part of the paraboloid $z=x^2+y^2$ that's inside the cylinder $x^2+y^2= 1$ with an outward ...
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Does the Gauss-Bonnet theorem apply to non-orientable surfaces?

I found statements of the Gauss-Bonnet theorem here, here, here, here, here, here, here, and here. None of them require that the surface be orientable. However, Ted Shifrin claims in a comment to this ...
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$df_x$ maps $N(S.X)$ isomorphically

This is actually just a linear algebra problem, but this text is taken from Guillemin and Pollack, Differential Topology on page 100. Real Problem: Let $f: X \to Y$ be smooth, $S = f^{-1}(Z)$ where ...
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1answer
61 views

Conditions for Matrix to be Product of Near-Identity Matrices

For $\epsilon > 0$, let $M_{\epsilon}$ be the family of $n$ x $n$ real matrices A such that $||$A$ - $I$_n|| < \epsilon$, where $|| \cdot ||$ is the standard operator norm. If $\epsilon$ is ...
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1answer
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If an orientation of a tree graph has no source vertices, must the in-degree of each vertex in said orientation be equal to one?

Given any polytree $T$ (any orientation of a tree graph) such that $\forall v\in V(T)(\text{indeg}(v)\neq 0)$ does this imply that $\forall v\in V(T)(\text{indeg}(v)=1)$? I'm pretty sure its true, but ...
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Maps between equivalence classes of vector bundles with and without orientation fixed at a point

I'm reading Hatcher's "Vector Bundles and K-Theory" and he says something that seems to be false. Here is a screenshot of the part of the book in question. He says that if we compare $Vect^n_0(S^k)$ ...
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Exponential map is orientation preserving

Let us define the map $f:\mathbb{R}\to \mathbb{S}^1$ by $x\to e^{2\pi i x}$. I want to prove that this map is an orientation-preserving map and also it's restriction on the unit interval has an ...
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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
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Relative orientation of satellite to a rotating plane.

We know the GPS position of a Geostationry satellite and a sensor. We also know the sensor orientation relative to the magnetic North of Earth. How do we find the orientation of Satellite relative to ...
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$M$ is orientable if and only if $M \setminus \{p\}$ is orientable.

Let $M$ a manifold of class $C^{\infty}$. Show that $M$ is orientable if and only if $M \setminus \{p\}$ is orientable. Comments: ($\Rightarrow$) Let $\omega: M \longrightarrow \Lambda^n(M)$ a ...
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1answer
75 views

Direction of path depends on sign of determinant of Jacobian

Let $f: K_1(0) \rightarrow \mathbb{R}^2$ be continuously differentiable, $\{z_1,..,z_m\}=f^{-1}(a)$ with a regular $a \in \mathbb{R}^2$. We choose $\epsilon$ small enough so that $f\vert_{\overline{U}...
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1answer
48 views

(Negative) Gradient and Orientability of its flow.

Before asking my question, I put the necessary definitions and some context. If you are used with Morse Theory, you can skip the text within [[[...]]]. [[[Let me first define what I mean by gradient ...
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1answer
36 views

Prove a DAG can be obtained by an undirected graph's longest cycle

Let G(V, E) be a finite undirected graph and let κ be its longest undirected cycle. Prove that it is always possible to obtain an orientation ω(κ) in which κ is topologically sorted and hence its ...
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1answer
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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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Let $M$ is not orientable and $p \in M$, then $M - \{p\}$ is orientable.

Is it true or false? Let $M$ a manifold of class $C^{\infty}$ not orientable and $p \in M$, then $M - \{p\}$ is orientable. I believe this to be false, but I do not know a counterexample
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Orientation of points in cell complexes

On page 268 - 269, Hatcher defines the notion of an orientation of a cell in his book on algebraic topology. He writes that for a cell complex $X$ with skeleton filtration $\varnothing =: X^{-1} \...
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Apllying Generalized Stokes Theorem

I'm having no difficulties to understand the theorical concepts about the Generalized Stokes Theorem, but I'm having trouble to applying it. Let $M$ the semiellipsoid in $\mathbb{R}^{3}$ defined by $...
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How to find orientation of cube?

I have two planes given in Hessian normal form. Now these two planes belong to a cube and I would like to find the orientation of that cube given the two planes. Intuitively it is clear to me that ...
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1answer
42 views

Continuity of the pointwise orientation on a manifold

I want to know how to define the continuity of a pointwise orientation on a manifold in a specific way (if this is possible), analogously to the continuity of a vector field as a continuous map from ...
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1answer
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Help to correct answer: Product of orientable manifolds is orientable

In the first reply to this post, I made a comment, but nobody answered me yet. Link: Orientability of a product of smooth manifolds implies orientability of each factor My problem is in the return of ...
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1answer
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What's the group with 2 elements: one for even permutations and one for odd permutations (on $n$ points)? How to construct it?

I was thinking about orientations (of a vector space, of a simplex, ...) and how there seem to always be exactly two orientations. Moreover, orientations are defined "up to the parity of the ...
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1answer
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How to define orientation mapping for parametrization of a 1-manifold in a 1 dimensional space

Here is a simple example that confuses me regarding the orientation mappings: In order to compute $\int_4^9{tdt}$ using $t=x^2$ parametrization in the interval $[-2,3]$ (i.e. $\int_{-2}^{3}{2x^3dx}$),...
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1answer
41 views

Working with the homological definition of orientation

I'm trying to understand the homological definition of orientation given by a continuous choice of generators for local homology groups. Unfortunately, I am already unsure about the proof $\mathbb R ^...
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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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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 ...