Questions tagged [orientation]

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

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Orientability issue when classifying symplectic structures on compact orientable surfaces

I have seen this post, where they identify symplectic structures on a compact orientable surface with $\mathbb R\smallsetminus0$, which I understand except for one issue regarding the orientation on $\...
Chris's user avatar
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$z \mapsto z^m$ is orientation preserving and regular

I'm following GP's book of Differential Topology and I stuck in the proof of Fundamental Theorem of Algebra using Intersection Theory. We have the afirmation of $f: \mathbb{S^1} \to \mathbb{S}^1, s.t.,...
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Proof construction for non-orientability of Mobius strip

I'm following Spivak's comprehensive guide to differential geometry and I'm getting a bit stuck on a calculation about orientability. The example is showing that the Mobius strip considered as a line ...
Bedge's user avatar
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need a rigorous proof of the generator of local homology group $H_{n}(\mathbb{R}^n,\mathbb{R}^n-\{0\})$ is the given simplex

My question: Why the generator of local homology group $H_{n}(\mathbb{R}^n,\mathbb{R}^n-\{0\})$ is that simplex $\sigma$, where $\sigma=[v_0,e_1,\cdots,e_n]$, $v_0=(-1,-1,\cdots,-1)\in \mathbb{R}^{n}$ ...
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What are oriented embeddings of zero-dimensional manifolds?

I'm trying to understand what it should mean for a submanifold of an oriented manifold to have a compatible orientation. For $1$-dimensional manifolds embedding into a $2$-dimensional manifold, I ...
Dylan Braithwaite's user avatar
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What is meant by a unit outer normal along an oriented hypersurface in an oriented Riemannian manifold?

I know that given a manifold $M$ with boundary ($\partial M\neq\emptyset$) and a point $p\in\partial M$, the vectors in $T_p M$ can be classified into three types: those which are also in $T_p(\...
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preimage of oriented submanifold under transversal map is orientable

Consider the map $f: M \to N$, transversal to the regular submanifold $S \subset N$. I.E. $df (T_x M) \oplus T_{f(x)} S = T_{f(x)} N$. We know that $f^{-1} S$ is a regular submanifold. Also we know ...
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Top homology $\mathbb{Z}$ implies orientability

Let $M$ a compact, connected $m$-manifold. Assume that $H_m(M;\mathbb{Z})\cong \mathbb{Z}.$ I want to show that then $M$ is orientable. My ideas: Let $o_M$ one of the two generators of $H_m(M;\mathbb{...
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Global orientation class of $M$ implies top homology is $\mathbb{Z}$

Let $M$ a compact, connected $m$-manifold. Assume that there exists a global orientation class, i.e. an $\sigma_M \in H_m(M;\mathbb{Z})$ such that for all $x\in M$ the element $(\rho_{x,M})_{\ast}(\...
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Integration over a closed surface in a Riemannian $3$-manifold

Let $\Sigma$ be a closed surface in a Riemannian $3$-manifold $(M,g)$. I was thinking about the validity of writing $$\int_\Sigma H^2 d\mu_\Sigma,\tag{1}$$ where $H$ is the mean curvature of $\Sigma$ ...
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$\mathbb Z_3$-orientable manifold is $\mathbb Z$-orientable.

Let $M$ be a $\mathbb Z_3$-orientable $n$-manifold. The goal is to show that $M$ is $\mathbb Z$-orientable. Assume that $M$ is closed and connected. Suppose, for a contradiction, that $M$ is not $\...
Luke's user avatar
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Orientation reversing diffeomorphism but no isometry?

Is it possible that an oriented Riemannian manifold $(M,g)$ with a large isometry group $\text{Isom}(M,g)$ has an orientation reversing self-diffeomorphism but no orientation reversing self-isometry, ...
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Consistency between two different ways of defining an orientation on $S^n$

In the book Topology, Geometry and Gauge Fields Interactions by Gregory L. Naber, he defines the standard orientation of $S^n$ (viewed as a subset of $\mathbb{R}^{n+1}$), $n\geq 2$ by the oriented ...
pofu curj's user avatar
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Non-vanishing of mod 2 intersection number for odd-dimensional manifolds implies the ambient manifold is nonorientable.

Let $X \subset Y$ be a compact embedded manifold of odd dimension such that $\dim X = \frac12 \dim Y = n$. At the end of the section titled "Oriented Intersection Theory", Guilleman and ...
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If $M$ is an oriented manifold with boundary, then this induces an orientation on $\partial M$.

Setup: Let $M^n$ be an oriented smooth manifold with boundary. We want to show that this induces an orientation on $\partial M$. Proof: Let $\mathcal{A} := \{(U_i,\varphi_i):i \in I\}$ be an atlas of $...
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Euler angles to ENU coordinates

Given the LLA coordinates and Euler angles (orientation) of a phone, where alpha = beta = gamma = 0 when the top of the phone is pointing north and facing up, I would like to find the unit ENU ...
Gen's user avatar
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Issue about the definition on orientation on manifolds

In my class, orientation on a $C^\infty$ manifold is defined as follows: Let $\mathcal{O}_0$ be the canonical orientation on $\mathbb{R}^m$ (i.e. the equivalence class (with respect to orientation) ...
Luigi Traino's user avatar
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Orientation of Dehn surgery manifold

Suppose we have a 3-manifold $M$ obtained by Dehn surgery along a given framed link on $S^3$. Then it has a natural orientation which comes from the standard orientation of $S^3$. Is it true that $-M$ ...
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Planar orientation density function in 3D

Let's say I have an assembly of randomly aligned cylinders. The orientation is presented with orientation density function (ODF) $f(\theta, \phi)$, where $\theta$ is the polar angle (w.r.t. $z$-axis) ...
user3563929's user avatar
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Stiefel-Whitney classes in Čech cohomology

Let M be a smooth manifold and Let G be a Lie group. We have the sheaf $\mathcal{F}_G=(F,\rho)$ of groups defined by $$F(U)=C^{\infty}(U,G)$$ for open sets $U\subset M$ (in particular set $F(\emptyset)...
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Orientation preserving isometries of the sphere

I know that the isometries of $\mathbb{S}^n\subseteq\mathbb{E}^{n+1}$ are $O(n+1)$. Intuitively, I think that $SO(n+1)$ should be the orientation-preserving ones. But I'd like to prove this rigourosly....
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Induced orientation given a short exact sequence (in Morse homology over $\mathbb{Z}$)

I am new to the field of algebraic topology and am currently studying Morse homology for a project. I read that given a short exact sequence $$ 0 \to A \to B \to C \to 0 $$ if we know the orientation ...
David's user avatar
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Orientation preserving parametrisation of torus

EDIT: After having posted the question along with an attempt at a solution, I could spot the mistake in my solution (it's easier to read LaTeX than your own imperfect handwriting...), so now it is ...
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Interpreting the Sign of the Jacobian Determinant

The sign of the Jacobian determinant of a two-dimensional transformation tells us if the transformation is locally orientation-preserving (if it's positive) or locally orientation-reversing (if it's ...
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Orientation bundle and its flat connection

Let $M$ be a smooth $n$-manifold (which is not assumed to be orientable), and write $o(TM)\to M$ for its orientation bundle. Equivalently, it is the top exterior bundle $\Lambda^n(TM)\to M$. In any ...
Ho Man-Ho's user avatar
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Why is the orientable double cover orientable?

According to the above definition of a global orientation, don't we have to show that both $\tilde\mu_{x}, \tilde\mu_{y}$ in $H_n(\tilde M, \tilde M \backslash \mu_x )$ and $H_n(\tilde M, \tilde M \...
zhongyuan chen's user avatar
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Orientation of a surface in the direction of an axis

I'm reviewing some vector calculus material in Chapter 16 of Stewart's Calculus (9th ed) and I want to make sure I understand what is meant by a surface being oriented in the direction of the positive ...
Leonidas's user avatar
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Proving $\mathbb{R}P^n$ is orientable if and only if $n$ is odd.

Prove that $\mathbb{R}P^n$ is orientable if and only if $n$ is odd. I know this question has been asked many times on this site, but all solutions consist of $n$ forms or homology groups which I can'...
Math101's user avatar
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Generators for de Rham cohomology on orientable products

I am trying to figure out how the orientation forms on $S^2$ constitute as generators for $H^2_{dR}(S^2 \times S^2)$. I've calculated that $H^2_{dR}(S^2 \times S^2) = \mathbb{R} \oplus \mathbb{R} = H^...
Lee Kwang's user avatar
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Show that $i^*\omega \in \Omega^{n}(S^n)$ is an orientation form on $S^n$.

Show that $i^*\omega \in \Omega^{n}(S^n)$ is an orientation form on $S^n$. Here $i:S^n \to \mathbb{R}^{n+1} - \{0\}$ and $\omega$ is a $n$-form on $\mathbb{R}^{n+1}- \{0\}$ for which $\omega_p(v_1, \...
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Is there a natural way to build a volume form on an oriented Poisson manifold?

Editet question Let $(M, \pi)$ be an oriented Poisson manifold. Is there a natural way to build a volume form from the Poisson bivector $\pi$? Original question It is always possible to build a volume ...
DavideL's user avatar
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Show that $\mathbb RP^n$ with the standard smooth structure is orientable if and only if $n$ is odd

Show that $\mathbb RP^n$ with the standard smooth structure is orientable if and only if $n$ is odd. I tried to show that the Jacobian of the transition map $$(u_1,u_2,\dots ,u_n)\mapsto \left(\frac{...
Sayan Dutta's user avatar
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Local diffeomorphisms and orientability

Given a differentiable map $f \colon S_1 \to S_2$ between regular surfaces such that $df_p$ is an isomorphism for each $p \in S_1$, I want to study whether the following statements are true or false: ...
David's user avatar
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Do non-closed surfaces have a canonical choice of normal vector?

For any orientable surface in $\mathbb{R}^3$, there are two possible choices for the normal vector at any given point. When the surface is closed, we can name those choices and differentiate between ...
TheAssistant's user avatar
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In the proof of the Stokes's theorem, why $\varphi |_{U \cap \partial M}^{\varphi(U)\cap \partial \mathbb{H}^n}$ is an orientation-preserving?

I am reading John Lee's Introduction to smooth manifold, second edition, proof of Theorem 16.11 and stuck at understanding some statement. Theorem 16.11 (Stokes's Theorem). Let $M$ be an oriented ...
Plantation's user avatar
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Standard Orientation on Lie Groups

I know that every Lie group is orientable. I want to know whether there is a standard choice of orientation. The standard orientation on $\mathbb{R}^n$ is $\left( {\partial \over \partial x_1}, \ldots,...
D.L's user avatar
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Every nonzero alternating $n$-tensor on a $n$-dimensional vector space $V$ determines an orientation of $V$

Proposition. Let $V$ be a vector space of positive dimension $n$. Then every nonzero alternating $n$-tensor $\omega$ on $V$ determines an orientation of $V$ by collecting all the bases $\{v_1,\ldots,...
Boar's user avatar
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Boundary Orientation

Edits to the original post are in bold. I've been going through @Ted Shifrin's lectures on Stokes's Theorem, and I had a question relating to his choice of orientation of the tangent space as it comes ...
RHyp's user avatar
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SO(3) derivatives in the paper "A Primer on the Differential Calculus of 3D Orientations"

In the paper A Primer on the Differential Calculus of 3D Orientations, the derivative of a SO$(3)$ member in equation (24) is given as a vector defined as $$\frac{\partial}{\partial t}\left(\Phi_{\...
Xiaohan Tang's user avatar
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Can a purely mathematical version of the right hand rule be given?

The right hand rule is a common convention for describing orientation of coordinates, used throughout physics. It's also used in the definition of the cross product. Is it possible to give a purely ...
SRobertJames's user avatar
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Stein and Shakarchi: "positive counterclockwise orientation"

I am having trouble understanding a passage in Stein and Shakarchi's complex analysis textbook. Here is the passage for reference. (It earlier wrote $z = re^{i \theta}$ and then wrote Euler's identity....
Mathematical Endeavors's user avatar
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Normal vectors of a piecewise smooth Jordan curve always point into the bounded Jordan domain.

It is a common definition to say that a smooth simple closed curve $\gamma:[0,1]\rightarrow\mathbb{C}$ is "positively oriented" if the normal vector at every point along $\gamma$ points into ...
Nick Mendler's user avatar
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Vector fields that make up a local frame are continuous

In Lee's book Introduction to Smooth Manifolds, on the section on orientations, he notes that Recall that by definition the vector fields that make up a local frame are continuous which I did not ...
CBBAM's user avatar
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How are transition functions of $TM $ defined when $ M$ is an oriented manifold?

This question was asked in my quiz of Differential geometry course and I am having a really hard time in this course. Question: Let $M$ be an oriented manifold of dimension $n$ and let $π : TM \to M$ ...
Jack's user avatar
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In proof of the Stokes's Theorem for Chains ( John Lee's Smooth manifolds )

I am reading the John Lee's Introduction to smooth manifold, second edition, Theorem 18.12 (Stokes's Theorem for Chains) and stuck at understanding some argument : I'm struggling with hours to ...
Plantation's user avatar
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3 votes
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Group of Oriented Edges of a Tiling

This is an idea I'm sure exists already, but is quite complicated, so it's hard to find without appropriate terminology. We consider a tiling of the plane by regular $n$-gons, containing an edge $e_0$ ...
Thomas Anton's user avatar
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Characteristic Property of the Orientation Covering (Problem 15-10 in Lee's Smooth Manifolds book)

The problem statement: Let $M$ be a connected nonorientable smooth manifold with or without boundary, and let $\widehat{\pi}: \widehat{M} \to M$ be its orientation covering. Prove that if $X$ is any ...
Tob Ernack's user avatar
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Effect on transition functions when changing the orientation of a manifold

Let $M$ be a smooth $n$-manifold. If an orientation on $M$ is given, then we can find an open cover $\{U_\alpha\}$ of $M$, and transition functions $g_{\alpha\beta}:U_{\alpha}\cap U_\beta \to \text{SO}...
blancket's user avatar
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Quaternion - calibration sensor [closed]

I have attached my sensor to a box. What I am interested in is the rotation of the box with respect to the world. In my case, to avoid the gimball lock, and beacause the sensor already provides it, I ...
cobdmg's user avatar
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Integrating a 2-form over 2-manifold with an orientation

This excercise asks me to integrate $\omega=z dx \wedge dy$ over the 2-manifold $A = \{(x, y, z): x^2+z^2=y, y<4\}$ with orientation $o(x)$ such that $o^{31}(x)>0$ using Divergence's theorem. So ...
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