The Stack Overflow podcast is back! Listen to an interview with our new CEO.

Questions tagged [orientation]

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

319 questions
Filter by
Sorted by
Tagged with
20 views

Orientation-preserving local difeomorphism

I was studying some notes of classes and I am stucked at the following proposition: Let $S_1,S_2,S_3$ be orientated surfaces. If $S_1\stackrel{f}{\to}S_2\stackrel{g}{\to}S_3$ are local ...
29 views
+50

Two connex components with different orientations - The surface cannot be orientable!

I need to prove that if $S$ is a surface covered by two coordinate neighborhood $V,U$ s.t. $V\cap U$ has two connex and the jacobian of coordinate change is positive in once and negative in other, ...
24 views

If exists two-form not null in $S$, so $S$ is orientable

I need to prove that, being $S$ a regular surface, If exists a two-form not null in $S$, so $S$ is orientable. Here, I need to use estricly the basic definition for orientability, ie, any two ...
19 views

Orientation of a curve in $\mathbb{R^3}$

Let $\gamma: \mathbb{R} \to \mathbb{R^3}, \gamma(t) := (0,0,1)+\cos(t)v_1+\sin(t)v_2$, where $v_1 =(\frac{2}{3},\frac{1}{3},-\frac{2}{3}),v_2=(\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}})$. I know this ...
43 views

Rigorous definition of left hand coordinate system.

It seems that for a 3D vector space over $\mathbb{R}$, we cannot define whether a coordinate system if left-handed or right-handed, since we cannot compare an abstract vector with our fingures. So how ...
47 views

Cross product definition with confusion around handedness

I want to check if my understanding about the cross product is correct. Wikipedia page on cross product says the definition of cross product depends on the orientation of the vector space. ...
21 views

Sign independency when rotating an inertia tensor with a rotation matrix

I have a set of points that represent a rigid cylinder for which I calculate its rotation in space (x,y,z) using the formula $\mathbf{J} = \mathbf{R^T} \mathbf{J_0} \mathbf{R}$ with R being the ...
18 views

Definition of orientation of curve using implicit function theorem

I am reading Henri Cartan's Elementary Theory of Analytic Functions of One or Several Complex Variables. On p.64, sec. II.1.9, he tried to show that if $\gamma=(\gamma_1,\gamma_2):[a,b]\to\mathbb R^2$ ...
23 views

Connected intersection of a manifold and orientation

From Do Carmo's book (Riemannian Geometry, P. 19) If M can be covered by two coordinate neighborhoods $V_1$ and $V_2$ in such a way that the intersection $V_1\cap V_2$ is connected, then $M$ is ...
10 views

XYZ position with orientation and altitude?

I am working on a project with sensors: I am able to have the orientation output of an object as quaternion, rotation matrix or euler angle. A second sensor is measuring the change is altitude of ...
23 views

Orientation for a vector spaces determines a canonical orientation for the dual vector space

Suppose $V$ is an $n$-dimensional real vector space, with $n > 0$. Show that an orientation for $V$ determines a canonical orientation for $V^*$, the dual of $V$. The idea I had in mind to show ...
31 views

Non orientable normal bundle gives a non simply connected manifold [closed]

Let $X$ be a compact connected manifold and $M\subset X$ be a compact connected hypersurface. If the normal bundle $NM$ is not orientable, then $\pi_1(X)\not= 0$.
24 views

Chain rule for differential calculus of 3D rotations

In the paper "A Primer on the Differential Calculus of 3D orientations" by Bloesch et al. they give an example on how to apply the identities derived for the derivatives of various rotation ...
86 views

Volume element of Haar measure on SO(3) with Euler angle parametrization

I have a real-valued function $f(\alpha, \beta, \gamma)$ which takes Euler angles $\alpha, \beta, \gamma$ as input, that I would like to average over the uniform distribution on orientations of 3-...
66 views

Seifert surfaces for knots $6_1, 6_2, 6_3$.

I have been trying to calculate the genera of these knots, but the first step in doing so is to convert them into orientable knots by constructing Seifert surfaces for those knots. I started to do ...
66 views

Confusion about the top homology group of a compact manifold.

I know that if the manifold is compact, then all of its homology groups are finitely generated. But on the other hand, we know (for example Hatcher 3.26) that if the manifold is closed and orientable, ...
57 views

Is orientation of Stokes theorem a convention?

Is orientation of Stokes theorem, that is, right hand rule, a convention? Can we also choose the left hand rule? But will not it create problems in Physics where the sign of our physical quantity (...
86 views

Position and orientation of local coordinate system relative to another local coordinate system (both described in world coord. system)

This question might have been answered before, but I couldn't find one that will best describe my problem, or before I got confused. The following image depicts three coordinate systems: a world ...
67 views

68 views

Is a smooth immersion $c: [a,b] \to M$ injective if its image is a 1-manifold with non-empty boundary?

My book is An Introduction to Manifolds by Loring W. Tu. Pictured below is the last example from Section 22, Manifolds with Boundary. In this question, it is confirmed that the example is an error. ...
56 views

What is the induced orientation on a 1-manifold with boundary that is the image of closed interval under these maps?

My book is An Introduction to Manifolds by Loring W. Tu. Pictured below is the last example from Section 22, Manifolds with Boundary. In this question, Prof Jack Lee says that the example is ...
30 views

Getting the orientation for a torus from the parametrisation

Say I have the torus $T$ in $\Bbb R^4$ given by $x^2+y^2=z^2+w^2=1$ parametrised (almost everywhere) by $$\Phi:(0,2\pi)\times (0,2\pi)\to \Bbb R^4$$$$\Phi(u,v)=(\cos u,\sin u,\cos v,\sin v)$$ Then I ...
20 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 ...
44 views

Orientation for compact manifolds

Page 253, -8 to -4 line Hatcher defines orientation for compact manifolds with boundary. A compact manifold is $R$_orientable if $M-\partial M$ is $R$-orientable. If $\partial M \times [0,1)$...
43 views

59 views

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 ...
24 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. ...
68 views

Orientation on manifolds

I am trying to understand the definitions here. In many books (say Tu or even Guillemin-Pollack) an orientation on a manifold is an assignment to affix $+1$ and $-1$ to classes of (tangent) basis. It ...
128 views

176 views

Every connected orientable smooth manifold has exactly two orientations, Lee Proposition 15.9

The proof of Proposition 15.9 from John Lee's book "Introduction to Smooth Manifolds" is left as an exercise. Here is the statement: Let $M$ be a connected, orientable, smooth manifold with or ...
216 views

What is the induced orientation on a 1-manifold with boundary that is the image of closed interval under a smooth immersion?

My book is An Introduction to Manifolds by Loring W. Tu. Pictured below is the last example from Section 22, Manifolds with Boundary. I have been trying to wrap my head around this for about 2 hours (...
112 views

Rotation with roll pitch and yaw in different coordinate system

Say I am given a point in an x1,y1,z1 coordinate system. I have a different coordinate system, x2,y2,z2 that has the same origin as the x1,y1,z1 system, but the axis are not aligned. I have roll, ...
29 views

Torque required to achieve a desired quaternion

I was hoping someone can either explain or direct me towards a source that can help me with the following problem (not for homework, more of a hobby). Given an object with a current quaternion $q_c$ ...
98 views

Diffeomorphism Between Surfaces Preserves Orientability

From Do Carmo (Exercise 2.6.2). Let $S_2$ be an orientable regular surface and $\varphi:S_1\rightarrow S_2$ be a local diffeomorphism at every $p\in S_1$. Prove $S_1$ is orientable. Up until this ...
I'm reading Warner. "Foundations of Differentiable Manifolds and Lie Groups." p. 138. I don't get the statement in the definition of orientable manifolds. 4.1 Definitions $\;$ (the preface omitted) ...