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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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On the unit sphere $S^2$, show the antipodal map $A:S^2\to S^2$ is orientation reversing using definitions.

I want to show that on the unit sphere $S^2$, the antipodal map $A:S^2\to S^2$ given by $(x,y,z) \mapsto (-x,-y,-z)$ is orientation reversing. I know that $S^2$ is a regular connected orientable ...
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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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$M$ 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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Orientation in five dimensions

I was wondering what the orientation of axes would be in five dimensions. In 3D, a right-handed orientation means $\vec{x}\times \vec{y}=\vec{z},\space\vec{y}\times \vec{z}=\vec{x},\space\vec{z}\...
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388 views

General definition of orientation-preserving (continuous) map of surfaces

Everybody seems to just use these maps but there is never a formal definition given. I looked at this question: Orientation preserving homeomorphisms but no answer is given and I can't make sense of ...
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The cofiber sequence $\mathbb{C}\mathrm{P}^{n-1}\to\mathbb{C}\mathrm{P}^n\to S^{2n}$

$\newcommand{\CP}{\mathbb{C}\mathrm{P}}\newcommand{\Z}{\mathbb{Z}}$(Let the notation $H^*(X)$ represent $H^*(X;\Z)$, i.e. ordinary unreduced cohomology with $\Z$ coefficients.) The standard inclusion ...
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An elementary but formal proof that the Moebius strip is not orientable

We define surfaces as images of $C^1$ functions from $K \subset\mathbb{R}^2 \rightarrow \mathbb{R}^3$, with $K$ compact, and we say a surface is orientable is we can pick continuously a normal vector ...
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187 views

$\Phi$ and $\Psi$ have the same orientation, prove there are at least two equivalence classes.

Fix a set $W \subseteq \mathbb R^n$ and let $k \leq n$. Define $S_W := \{ \Phi:U \to W: U \subseteq \mathbb R^k $ is open, $\Phi $is a smooth embedding, and$ \Phi(U)=W \}$. Suppose that $...
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Showing that the orientable double covering of a manifold is an orientable manifold (do Carmo Differential Forms ex. 3.16)

I am trying to solve exercise 3.16 of do Carmo's Differential Forms and Applications. The problem is as follows: Let $M$ be a connected differentiable manifold. For each $p\in M$, denote by $\...
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Homology orientation induced by manifolds

To define Seiberg-Witten invariants one needs homology orientation. So for a closed oriented smooth (let us say as well simply connected) 4-manifold $M$, a homology orientation is an orientation of $...
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Orientable cover of a non-orientable manifold factored through the orientation double cover.

While proving that orientable cover $M$ of a manifold non-orientable manifold $N$ factored through the orientation double cover, I got stuck in this following problem... If $p:N→M$ is a covering, N ...
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Definition of Integration of a Differential From in Lee's Introduction to Smooth Manifolds.

On pg. 402 of Lee's Introduction to Smooth Manifolds (Second Edition), the following is said to define the integral of a differential form on $\mathbf R^n$: Let $D$ be an open domain of integration (...
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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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$R$ 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 ...
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Orientation and Coloring

Prove that the following two conclusion is equivalent for simple graph $G$. (i) The chromatic number $\chi (G) \leq k$; (ii) $G$ has an orientation where no directed path of length $k$ exists. ...
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Are tubular neighborhoods orientable?

Let $M$ be a smooth $m$-manifold and $N \subseteq M$ a smooth $n$-dimensional submanifold. Then there is a so called tubular neighborhood $(E,p)$ of $N$ in $M$, which is a neighborhood $E$ of $N$ in $...
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429 views

Parametrization of the Möbius strip and orientation

I'm supposed to prove that the Möbius strip is not orientable by using this lemma: Lemma: Let $S\subset\mathbb{R}^3$ be a surface such that $S=S_1\cup S_2$, where $S_1$, $S_2$ are connected, ...
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561 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 ...
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1answer
745 views

Contour integration and reversing orientation of parametrisation

I have been reading through "Complex Analysis for Mathematics & Engineering" by J. Matthews and R.Howell, and I'm a bit confused about the way in which they have parametrised the opposite ...
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1answer
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How does a Möbius strip not have an area?

The self contained version of this question is: "Given a Möbius strip $M$ do there exist two curves $\gamma_1,\gamma_2$ in $M$ such that $M\setminus\gamma_1$, $M\setminus\gamma_2$ are orientable and ...
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Show that $M=\{(x,y,u,v) \in \mathbb{R}^4 : x^2 + y^2 = u^2 + v^2 = 1\}$ is orientable, explaining the induced orientation.

Let $M=\{(x,y,u,v) \in \mathbb{R}^4 : x^2 + y^2 = u^2 + v^2 = 1\}$. Show that $M$ is an orientable subvariety of $\mathbb{R}^4$, explaining the induced orientation. Consider the $2$-form $\...
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$R$ orientation in Hatchers

The precise source is Chapter 3, page 235, - line 9. So what Hatcher has stated is: $M$ is an $n$ manifold, and $R$ is a ring, and $x \in M$. Then $H_n(M,M-x;R) \simeq H_n(M,M-x; \Bbb Z) \...
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Metric to describe the “distance” between and orientation and position pair.

I have two points and would like a metric that tells me how close those points are to each other. Each point is described by both a 3D position and a 3D orientation. I can determine the distance ...
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Orientation of a Manifold with Trivial Tangential Bundle

Let $M$ be a smooth (eg $C^{\infty}$) manifold. Let assume that $M$ has trivial, oriented tangent bundle $TM$, so $TM \cong M \times \mathbb{R}^n$ for appropriate $n$ and orientable. How to conclude ...
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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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2answers
85 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
123 views

Simple closed curve orientation definition

The (simple closed curved) curve istraversed counterclockwise, and said to be positively oriented, if the region it (what is "it")? encloses is always to the left of an object as it (what is "it"?) ...
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For path connected manifolds, the orientation double cover is path connected iff manifold is not orientable

The "orientation double cover" $\tilde{M}$ of a n-manifold $M$ is defined as follows: $\tilde{M} = \{\mu_x | x \in M, \mu_x \textrm{ is a generator of } H_n(M,M-\{x\})\}$. Well, if we assume that the ...
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342 views

Smooth atlas oriented iff transition maps have positive Jacobian determinant

First of all, my definition: An orientation of an $m$-dimensional smooth (connected) manifold $M$ is a choice of orientations for all tangent spaces $T_p M$, $p \in M$, such that there exists an atlas ...
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41 views

If $M^n$ is a submanifold of $\mathbb{R}^{n+k}$ such there are L.I. vector fields $v_1,\ldots,v_k$ such that $v_i \perp T_pM$ then $M$ is orientable

If $M^n$ is a submanifold of $\mathbb{R}^{n+k}$ such there are L.I. vector fields $v_1,\ldots,v_k$ such that $v_i \perp T_pM$ for every $p \in M$ then $M$ is orientable. My attempt is: Once $M^n$ ...
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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 ...
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1answer
284 views

Number of all possible orientations in a graph

What is the number of all possible orientations for an undirected graph? I think it must be $2^{|E|}$, because we have $|E|$ edges, each of them can have 2 choices for it's direction. Is it true?
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543 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
96 views

define orientation of quotient space

How can one expand the definition of orientation of vector spaces to corresponding quotient space? For example, if I am given two vector spaces V and W, and I know the definition of orientation for ...
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1answer
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Prove that $S$ is colorable if and only if it is orientable

I am taking a course on algebraic topology and I am trying to prove the following exercise: Let $S$ be a differentiable surface in $\mathbb{R}^3$. Prove that $S$ is colorable (you can paint one ...
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1answer
490 views

Orientability and Hypersurfaces

I got stucked in this problem: Show that: i) Every embedded closed hypersurface $S$ is orientable. ii) Every differentiable hypersurface defined by a regular cartesian equation $\ g(x_1,..., x_n)=0$ ...
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1answer
84 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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1answer
47 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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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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1answer
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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
84 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
89 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
83 views

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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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
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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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87 views

Orientations of pixels of image

I want to get an approximate orientation of each pixel of an image. I'm using C++ in combination of OpenCV (an open MatLab library) for this. My initial image is this: After some reading I found ...
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1answer
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First obstacle to triviality is orientability

I am reading chapter on Characteristic classes from the book Vector bundles and K-theory by Allen Hatcher. When giving motivation for what does characteristic classes measure author says that The ...
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2answers
244 views

Why boundaries oriented this way for Stokes's Theorem?

In the proof above, why do the boundary curves of $ S_1$. and $ S_2$ need to be oriented in opposite directions? Why are the boundary curves of each respective hemisphere oriented in opposite ...
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1answer
152 views

Moebius strip orientability

Let $\{M\}$ be the one-sided Moebius strip and $\{MM\}$ the corresponding two-sheeted Moebius strip. Let us assume $\{MM\}$ to be the doubling of $\{M\}$; hence it is orientable. Then, $\{M\}$ is ...
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1answer
46 views

How to show $(x\cdot y, x\cdot z) = ||x||^2 (y,z) \quad \forall x,y,z \in \mathbb{H}$

In the book of Linear Algebra by Werner Greub, at page 209, it is given that $$(x\cdot y, x\cdot z) = ||x||^2 (y,z) \quad \forall x,y,z \in \mathbb{H}$$ , which are easily verified using (7.36) ...