Questions tagged [orientation]

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

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Orientation of Set - Integration

This question is not splendid (I would say "stupid") but what why there is no definition for the orientation of a segment $[a, b]$? In multivariable calculus, there are manifolds and ...
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Why does it seem possible from Physics to say something about orientability locally?

In Quantum Field Theory, if I understand correctly, Physicists showed that certain process are not symmetric under time reversal. This should be a local thing, mathematically, and would show that ...
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Normal Bundle Orientable and inclusion null homotopic then M is orientable

I am trying to do the following exercise : Let $M$ be a submanifold of $N$, both without boundary. Show that if the normal bundle of $M$ in $N$ is orientable and $i: M\rightarrow N$ is null-homotopic ...
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On the “Non-Conservation of Parity in Weak Interactions”

Kostrikin and Manin, in their Linear Algebra and Geometry, state that: (The excerpt is on pp. 42-43.) The statement comes after a proof of general linear group over reals having two connected ...
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Problem of inward outward vector in degree $0$ Lemma by Milnor

In order to prove the Lemmma $1$ at pag $28$ of differential topology : Lemma : Let $\tilde{M}$ be a compact $(n+1)-$manifold, $M = \partial \tilde{M}$ with induced orientation, $M$ compact and $f : ...
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$\overline{B(p,r')} - B(p,r)$ is a $n$-manifold with $\partial M = \partial B(p,r) \cup \partial B(p,r')$

Given $X : U \subset \mathbb{R}^{n} \longmapsto \mathbb{R}^{n}$ smooth vector field on $U$ open in $\mathbb{R}^{n}$ we want to define the index $i_{X}(p)$ of a vector field in the point $p$ where $p$ ...
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Outward pointing vectors on the boundary of the interval $[a,b] \subset \mathbb{R}$

According to the Loring W. Tu's book "An Introduction to Manifolds", we have a standard oriented atlas on $[a,b]$, with the charts $([a,b[, \phi)$ and $(]a,b],\psi)$, where $\phi \colon [a,b[...
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Why do we need orientation to integrate on a manifold?

In my differential geometry course we are studying the integration of compactly supported $n$-forms and I do not really get why we need our manifold to be oriented. Indeed, we define the integral of ...
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A polynomial square root defined on a Lie subalgebra

" Let G be a connected compact Lie group, and H a connected Lie subgroup of G such that G and H have the same rank. We denote by $\mathfrak{g}$ and $\mathfrak{h}$ the respective Lie algebras. We ...
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How to carry around local frames in tangent space ? Degree $0$ Lemma

In a proof of the "Degree 0 Lemma" i.e. Lemma : Let $\tilde{M}$ be a compact $(n+1)-$manifold, $M = \partial \tilde{M}$ with induced orientation, $M$ compact and $f : M \longmapsto N$ of ...
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Orientation of boundary of cylinder $ M \times [0, 1] $

I have to prove that if $M$ is boundaryless smooth manifold and $M \times [0, 1] $ is oriented, then when we identify $M \times \lbrace 0 \rbrace$ and $M \times \lbrace 1 \rbrace$ with $M$, we have ...
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Problem with calculating relative orientation

I am using an IMU which provides absolute orientation of the sensors frame $S$ relative to an earth-fixed frame $N$ in quaternion form, $^S_Nq$. In my experiments, I first obtain an initial ...
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$\mathbb{R}P^3$ is orientable, but $\mathbb{R}P^2$ is not

Def: Let $U, V \subset \mathbb{R}^n$ be open and $f: U \to V$ a homeomorphism. We say $f$ is orientation preserving if for all $x \in U$ the composite $$ H_n(\mathbb{R}^n, \mathbb{R}^n \setminus 0) \...
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Oriented manifolds and the intermediate value theorem

Working through a proof that any connected orientable manifold has two orientations (From Tu's manifolds), I'm having a difficulty with a small nuance in the proof. Consider a manifold $ M $ with two ...
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Flaw in proof that tangent bundle is orientable

I'm trying to show that the manifold of a tangent bundle is orientable. However, I evidently made a mistake in my proof, and was wondering if someone could help me point out where I'm going wrong. I'm ...
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Determinant in calculation of orientation in Riemannian Geometry [closed]

Assuming not all of $y_1,..., y_n$ are zero. When $i\ne j$, define $$ c_{ij}= \frac{2y_iy_j}{\left(\sum_{k=1}^n y_k^2\right)^2} $$ and $$ c_{ii}=\frac{-\left(\sum_{k=1}^n y_k^2\right)+2y_i^2}{\left(\...
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Eigen vectos and Calculation of orientation angles from symmetric 2nd order orientation tensor

I have a list of the evolution of symmetric 2nd order orientation tensor $A_{ij}$ with time. ...
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1answer
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Rotating angular velocities

I have a quadcopter with yaw, pitch, roll and their respective derivatives p, q and r. ...
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How is the cross product an orientation dependend pseudo-vector, since the right hand rule uniquely determines its direction?

The geometric definition of the cross product $a\times b$ of two vectors (arrows) $a$ and $b$ seems unambiguous, in particular its direction is uniquely determined by the right hand rule. So why is ...
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Plane of symmetry and geodesics in surfaces

Let $(M,n)$ be an oriented regular surface of $\mathbb{R}^{3}$ such that the intersection of $M$ with the plane $\Pi:=\{(x,y,z) \in \mathbb{R}^{3}:z=0\}$ is the image of a curve parametrized by arc ...
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Sign of a classical field theory Lagrangian

In classical field theory the action of massive real scalar field on a curved background reads: $$I = \int_M \sqrt{-g}\,d^4x\,\left[ - \frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}...
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A smooth (n-1)-surface $S \subset \mathbb {R}^n$ is orientable $ \implies S $ has a continuous field of normal vectors to it.

I would like to be able to show that: if a (n-1)-surface $S \subset \mathbb {R}^n$ of class $ C ^ 1 $ is orientable $ \implies S $ has a continuous field of normal vectors to it. Below I show where I ...
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Conventions of Thom isomorphism and orientations

I was recently reading some texts on Thom classes and the Thom isomorphism (e.g. Differential Forms in Algebraic Topology by Bott and Tu). There it is stated that one of the defining characteristics ...
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Understanding 0-dim. case in proposition 15.3 in “Introduction to smooth manifolds” by Lee

I am trying to understand the proof of the following proposition. It is taken from Lee's book "Introduction to smooth manifolds": I have trouble understanding the 0-dim. case. In partucluar ...
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Notion of degree of a map from an orientable manifold to non-orientable manifold

Before writing my question, I want to write something that I know. Let $M$ and $N$ be two closed(compact, without boundary) connected topological manifolds of dimension $n$. Now, if both are $\Bbb Z$-...
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Have I really proved Exercise 23.5 in Tu's Manifolds, or how to amend this proof?

In the following proof, I believe I have only showed that the $2$-form given equals the orientation form $\omega$ on the intersection $U_x\cap U_y\cap U_z$. I'm not sure if I can do the same `trick' ...
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Degree of covering map $S^n \to \mathbb{R}\mathrm{P}^n$ and local degree signs

This question is both about a specific problem and a soft question about these sorts of problems. I'm reading through Hatcher's textbook and having trouble dealing with signs when it comes to local ...
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An exercise on orientation of surfaces

From Zorich II, pag. 177, exercise 4a: A subspace $\mathbb{R}^{n-1}$ has been fixed, a vector $\mathbf{v}\in\mathbb{R}^n\setminus\mathbb{R}^{n-1}$ has been chosen, along with two frames $\{\mathbf{\...
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Show that this diffeomorphism family $T_t$ is orientation-preserving for small enough $t$

Let $\tau>0$, $d\in\mathbb N$, $T_t$ be a $C^1$-diffeomorphism on $\mathbb R^d$ for $t\in[0,\tau]$ with $T_0=\operatorname{id}_{\mathbb R^d}$ and $\Omega\subseteq\mathbb R^d$. It's easy to see that ...
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Fixing orientation of connected smooth manifold in $\mathbb{R}^n$ by fixing a frame in any of its tangent spaces

I would like to show that fixing the orientation of $ k $ -manifold smooth connected $ S $ in $ \mathbb {R} ^ n $ is equivalent to fixing a frame for one of its tangent spaces. (Source: Zorich, ...
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meaning induced orientation on boundary manifolds

let $M$ one $k$- manifold orientable on $\mathbb R^n$ with boundary and $p\in \partial M$ and $\alpha$ one coordinate patch around $p$. i know that the induced orientation on $\partial M$ By Munkres (...
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Orientation of frames generated by consistent charts in a common point

I have a smooth $k$-surface $S\subset \mathbb{R}^n$ and two charts $\varphi_1:I_t^n\to U_1\subset S$, $\varphi_2:I_\tau^n\to U_2\subset S$ with $U_1\cap U_2\neq \emptyset$ ($I^n$ is the unit open cube ...
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Fixing orientation of connected smooth manifold in $\mathbb{R}^n$ by normal unit vector

My question is the natural continuation of this. In that topic I understood why, for a connected $ S \subset \mathbb {R} ^ n $ smooth manifold, fixing its orientation is equivalent to: show any chart;...
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Let F$(x,y,z)=(xy,yz,xz)$ and $S=\{(x,y,z):x^2+y^2\le1,0\le z\le1\}$. Compute $\iint_{\partial S}\text{F}\cdot \text{n}dA$

$\def\hl#1#2{\bbox[#1,1px]{#2}} \def\box#1#2#3#4#5{\color{#2}{\bbox[0px, border: 2px solid #2]{\hl{#3}{\color{white}{\color{#3}{\boxed{\underline{\large\color{#1}{\text{#4}}}\\\color{#1}{#5}\\}}}}}}} \...
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Fixing orientation of connected smooth manifold in $\mathbb{R}^n$ by a single chart

I am studying on Zorich, Mathematical Analysis II, 1st ed. pag. 174-175. After having properly explained how orientations (equivalence classes) are defined for smooth k-dimensional surfaces in $\...
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Doubts about the Zorich explanation of coordinate systems and orientation classes

I make a minimum of preamble by reporting what is written in Zorich, Mathematical Analysis II, 1st ed., Page 170-172: Let $G$ and $D$ be diffeomorphic domains lying in two copies of the space $\...
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Tangent bundle $TM\to M$ is an orientable bundle iff $M$ is orientable

This is Example 6.3 in Bott-Tu, which asserts a smooth manifold $M$ is orientable iff the tangent bundle $TM\to M$ is an orientable bundle. If $A=\{(U_\alpha,\psi_\alpha)\}$ is an atlas for $M$, then ...
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Expressing a differential form orienting a manifold

For reference this questions pertains to Adams and Essex, Calculus 2, chapter 17.4, Example 2. From the book (slightly abbreviated): A smooth (n-1)-manifold $\cal{M}$ in $\mathbb{R}^n$ is orientable ...
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Nonzero class in integral homology implies orientability

Let $M^3$ be a compact, connected and oriented $3$-manifold with nonempty boundary and let $\Sigma^2$ be a compact and embedded surface such that $\Sigma \cap \partial M = \partial \Sigma$. If $\Sigma$...
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Orientation of the product bundle of two oriented vector bundles

I am reading "Characteristic Classes" of Milnor and Stasheff. Let $\xi=(\pi:E\to B)$ be a real vector bundle of rank $n$. An orientation for $\xi$ is a function which assigns an orientation ...
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If $(M,g)$ is a Riemannian manifold and $S$ is a regular level set of $f:M\to \Bbb R$ then $\text{grad}f|_S$ is nowhere vanishing

I have a question reading a proof of the following theorem. Theorem. Let $M$ be an oriented smooth manifold, and suppose $S\subset M$ us a regular level set of a smooth function $f:M\to \Bbb R$. Then $...
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orientable manifold with $k$-connected component.

Find how many possible orientation there are for an orientable Manifold with $k$-connected component. Can we say that , there are $2^{k}$ possible orientation becasue for all connected component ...
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1answer
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$\omega$ a continuous $m$-form in $M$, $\dim M=m$, $M$ is orientable $\iff \omega \neq 0$

Let $M$ a surface, $\dim M=m$, and let $\omega$ a continuous $m-$form in $M$. I want to prove that $M$ is orientable $\iff \omega(x)\neq 0\; \forall x\in M$. The $(\Leftarrow)$ part is ok to me, ...
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Doubt about orientation of differential manifolds (using gradient)

In the lectures we defined a volume form on a differential manifold; in particular we explicited how to orient a $d$-dimensional submanifold of $\mathbb R^{d+c}$ (let say $M=f^{-1}(b)$, with $f\in C^\...
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Conjugate of quaternion doesn't give expected result

I'm using this site to play with quaternions. All of my quaternions are unit quaternions. I find quaternion of some Euler Angles(x, y, z) by using the website -inputs are degree and ZYX order Euler- ...
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Getting yaw, pitch or roll parts from a quaternion

I have a unit quaternion q = w + xi + yi + jk This quaternion means a rotation around an axis. I need to get/extract only one component of this rotation (only yaw, ...
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1answer
106 views

Isn't $[0,1]$ orientable?

It's not possible to put an atlas (of manifold with boundary) on $[0,1]$ because any chart whose domain contains $0$ has to have a positive derivative while any chart whose domain contains $1$ has to ...
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Orientability between diffeomorphic surfaces

From Do Carmo's Differential Geometry of Curves and Surfaces, 2nd edition, Now, when I tried to prove this, I reached this point: Up until this box, I couldn't understand the philosophy behind ...
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Determining that a certain diffeomorphism of $\Bbb R^n-\{0\}$ is orientation preserving or not

Consider the diffeomorphism $f:\Bbb R^n-\{0\} \to \Bbb R^n-\{0\}$ (whose inverse is itself) given by $x\mapsto x/|x|^2$. How can we determine that $f$ is orientation preserving? For $n=1$ it is ...
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Attempted “proof” of relationship between extrinsic and intrinsic rotation sequences

Here is my attempt at a "proof" that 3 intrinsic rotations in the sequence $x$, $y'$, $z''$ is the same as 3 extrinsic rotations but in the reverse order $z$, $y$, $x$. Suppose we have 3 intrinsic ...

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