Questions tagged [orientation]

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

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377 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 ...
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81 views

What would be the concept of “Infinite Dimension Orientability”?

We have the following definition of orientability for finite dimensional vector spaces: Definition: Let $\Bbb{E}$ be a finite dimension vector space and $\mathscr{B}(\Bbb{E})$ be the set of all ...
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137 views

A vector bundle which has an orientation-reversing isomorphism has a subbundle of rank $1$?

Let $E$ be a smooth real vector bundle of even rank, over a smooth manifold $M$. Suppose there exist an orientation-reversing vector bundle isomorphism $\Phi:E \to E$. Is it true that $E$ has a ...
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103 views

Why all the alternating signs?

For instance: $ |A \cup B \cup C|=(|A|+|B|+|C|)-(|A \cap B| + |A \cap C| + |B \cap C|)+|A \cap B \cap C|$ $\chi(X) = F- E + V = \sum_{i} (-1)^i \text{rank}(H_i(X)) =\sum_{i} (-1)^i \text{rank}(C_i(X))...
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64 views

Can the concept of orientability be applied to more general spaces?

Today a friend asked me if the Moebius strip with one segment identified to a point is orientable or not. The first thing I replied is that it is not a manifold so you can't define orientability in ...
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70 views

Topological invariants that detects change of orientation for an odd dimensional manifold.

Assume all manifolds involved to be closed and orientable. During my studies I learnt about the signature of a $4k$-manifold, and it turned out that (when it is non zero) it can be use to make a ...
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619 views

Product of Two Orientable Manifolds is Orientable

I am trying to show that following: Let $M$ be an oriented smooth manifold of dimension $m$, and $N$ be an oriented smooth manifold of dimension $n$. Then $M\times N$ is orientable. Let $\pi_M:M\...
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157 views

How to define orientation on infinite dimensional vector space

Let $\mathbb{V}$ be a real Banach space (if someone knows the answer for more arbitrary T.V.S. then great). Is there some concept of orientation that one could define on $\mathbb{V}$ that matches the ...
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148 views

Orientability of variety

Consider the real projective variety on $\mathbb{R}P^n \times \mathbb{R}P^n$ defined by the bihomogeneous polynomial $x_0y_0+...+x_ny_n$, i.e. $V=\{([x_0:...:x_n],[y_0:...:y_n])|x_0y_0+...+x_ny_n=0\}$....
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133 views

$\Phi:M\to M$ be a smooth map whose restriction to $\partial M$ is the identity

Let $M$ be a connected oriented smooth n-dimensional manifold-with-boundary, and let $\Phi:M\to M$ be a smooth map whose restriction to $\partial M$ is the identity. If $\omega$ is any smooth n-...
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272 views

Induced orientation on manifold

Suppose $M$ is a naturally oriented $n$-manifold embedded in $\mathbb{R}^n$ with non-empty boundary. Give $\partial M$ the induced orientation. The induced orientation defines a unit normal vector ...
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370 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 ...
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97 views

Calculating the size of Radial Shadow that M casts on a sphere centered at $(a,b,c)$ times $-4\pi$

I have a question about this problem and I want to understand it, but I'm not sure if my logic is solid. The question is: Let M be compact, connected, oriented surface with boundary in $\mathbb{R}^...
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125 views

Is any orientable smooth manifold of dimension $3$ with two independent vector fields parallelizable?

Let $M$ be a $3$ dimensional smooth manifold such that there exists two non vanishing independent vector fields $X_1, X_2 \in \mathfrak{X}(M)$. Given that $M$ is orientable, is $M$ also parallelizable?...
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154 views

Is the fixed point set of a self-diffeomorphism of odd order orientable?

Let $M$ be a smooth oriented manifold and let $f \colon M \to M$ be a self-diffeomorphism with $f^p = \text{id}_M$ for some odd $p > 0$. Then $f$ is orientation preserving and it follows from the ...
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133 views

Why is the preimage orientation given by a transversal map smooth?

On page 100 of Differential Topology, Guillemin & Pollack define, given a smooth map $f: X \rightarrow Y$ between an orientedmanifold with boundary and an oriented boundaryless manifold and a ...
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57 views

Proving that an $E$-oriented manifold has an $E$-oriented normal bundle

This is the setting we are working in: $M$ is a closed, smooth $n$-manifold embedded in $\mathbb{R}^{n+k}$ with a chosen embedding $e\colon M^n\to \mathbb{R}^{n+k}$. It is $E$-oriented, for $E$ a ...
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79 views

Describing the Unit Normal to a Cylinder in $\mathbb{R}^{3}$

This is problem 34.4 in Munkres' Analysis on Manifolds. Let $\mathcal{C} = \{ \ (x,y,z) \in \mathbb{R}^{3}\mid x^{2} + y^{2} = 1 \text{ and } 0 \leq z \leq 1 \ \}$. Orient $\mathcal{C}$ by declaring ...
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91 views

Compute the degree of $\phi:(x_1,x_2,x_3,x_4) \mapsto (x_1,-x_3,-x_2,x_4)$. Does $\phi$ preserve orientation?

Consider the map $\phi:S^3 \rightarrow S^3$ given by $\phi(x_1,x_2,x_3,x_4)=(x_1,-x_3,-x_2,x_4)$. Compute the degree of $\phi$. Does $\phi$ preserve orientation? First I want to point everything ...
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65 views

Relations between the homotopy class and the orientation of the connected sum of two manifolds

I want to find some general arguments about why it can happen that $$ M \sharp \overline{M} \not\simeq M \sharp M$$ for $M$ a compact orientable odd dimensional manifold, which is chiral, i.e. ($M \...
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110 views

Orientation at the boundary for manifold with corners: the simplex

Consider the $n$-simplex $$\Delta[n]:=\{(t_{1},\dots,t_{n})\in \mathbb{R}^{n}\: : \: 0\leq t_{1}\leq t_{2}\leq \dots \leq t_{n}\leq 1\}.$$ This is a manifold with corners. The cofaces map $d^{i}\: : \...
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104 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 ...
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17 views

How is a $k$-form integrated over an oriented smooth $n$-manifold in the case it is connected?

I have seen in several answers to questions on this page stating that there is no way to integrate a $k$-form over an oriented smooth $n$-manifold if $k \neq n$. However I cite Tu in his book on ...
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23 views

If $S \subset X$, with $\partial S = S \cap \partial X$. Must $T(S \cap \partial X) = T(S) \cap T(\partial X)$?

Some background. This came about in the proof about boundary orientation $$\partial f^{-1}Z = (-1)^{\operatorname{codim}Z} (\partial f)^{-1}Z.$$ A reference would be Guillemin-Pollack page 101. ...
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28 views

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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50 views

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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108 views

$M$ 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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393 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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60 views

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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173 views

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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297 views

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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82 views

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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288 views

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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40 views

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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30 views

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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41 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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100 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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0answers
18 views

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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0answers
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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0answers
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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0answers
179 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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0answers
57 views

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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22 views

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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0answers
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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0answers
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):=(\...