Questions tagged [orientation]
For question regarding the notion of orientation both in topology and in global analysis.
559
questions
1
vote
0
answers
13
views
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 ...
- 2,982
1
vote
0
answers
41
views
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 ...
- 537
1
vote
0
answers
28
views
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 \...
- 643
1
vote
0
answers
28
views
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 ...
- 1,141
4
votes
1
answer
89
views
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'...
- 3,837
2
votes
1
answer
45
views
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^...
- 23
1
vote
0
answers
31
views
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, \...
- 33
2
votes
1
answer
69
views
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 ...
- 416
1
vote
0
answers
69
views
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{...
- 7,792
0
votes
0
answers
30
views
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:
...
- 61
0
votes
1
answer
33
views
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 ...
- 109
1
vote
1
answer
107
views
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 ...
- 1,680
0
votes
0
answers
30
views
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,...
- 442
0
votes
1
answer
30
views
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,...
- 785
0
votes
1
answer
72
views
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 ...
- 83
0
votes
0
answers
68
views
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_{\...
0
votes
1
answer
109
views
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 ...
- 2,223
0
votes
0
answers
36
views
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....
1
vote
0
answers
36
views
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 ...
- 123
2
votes
1
answer
93
views
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 ...
- 3,345
0
votes
1
answer
67
views
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 $π : TM \to M$ ...
- 67
0
votes
0
answers
68
views
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 ...
- 1,680
3
votes
0
answers
45
views
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 $e_0$ ...
- 2,094
1
vote
1
answer
50
views
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 ...
- 4,149
0
votes
0
answers
25
views
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}...
- 2,016
0
votes
1
answer
44
views
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 ...
- 3
1
vote
0
answers
64
views
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 ...
- 33
1
vote
0
answers
42
views
Weak Parallelisability/Strong Orientability
An orientable manifold is defined by looking at small patches, looking at the interaction of local homology groups on these small patches, and focusing on when this gives us a coherent global story. ...
- 2,094
0
votes
0
answers
42
views
Oriented matroids: A map between chirotopes and the set of covectors?
I am just getting used to the topic of oriented matroids. I was surprised to read that a central aspect of oriented matroid theory is that there exists a "non-trivial equivalency" between ...
0
votes
0
answers
22
views
How to produce an N-dimensional *direct* basis?
I need a procedure to generate an N-dimensional orthonormal direct basis.
The 1D, 2D, and 3D case are not a problem. For example, in the 3D case, if I have 3 vectors $v1, v2, v3$ (suppose they behave ...
- 1
0
votes
0
answers
175
views
surface integrals orientation. when is it positive or negative
For this question below, I don't understand what they're asking. what does it mean to find the net flow rate upward through the surface? How do I know if I need to put a negative before computing the ...
- 751
0
votes
0
answers
15
views
Orientation of simplicial complexes [duplicate]
I'm trying to understand simplicial homology over $\mathbb Q$ and I'm having some problems in grasping with the definition of orientation of an abstract simplicial complex, the major one being that I ...
- 418
0
votes
1
answer
60
views
Degree of a map between orientable closed manifolds
Let $f:X\rightarrow Y$ be a local diffeomorphism, where $X^n$, $Y^m$ are compact and orientable manifolds. I'd like to show that $deg(f)=\text{#}(f^{-1}(z))$ or $deg(f)=-\text{#}(f^{-1}(z))$, for $z \...
- 436
0
votes
0
answers
22
views
Confused about "twist fluctuations" for knots
This question is referring to the following article
https://news.mit.edu/2020/model-how-strong-knot-0102
So this article says that a knot is stronger if it has more "twist fluctuations" ...
0
votes
0
answers
24
views
Compute orientation without a basis?
Follow up to this question (How to count polyhedral rotations?): If you have a regular polytope $P$ with graph $G_P$, you can easily use a computer to list all the graph automorphisms of $G_P$. Half ...
- 16.7k
0
votes
0
answers
12
views
Does the choice of orientations of $V$ and $W$ determine an orientation of $V\oplus W$?
Let $V$ and $W$ be two finite-dimensional real vector spaces. I guess that there is only one way to interpret the title (if my guess is wrong, then please let me know), but here is what I mean:
...
- 2,924
5
votes
1
answer
43
views
Show that contact manifolds are orientable, or why $\lambda\alpha\wedge(\lambda d\alpha+d\lambda\wedge\alpha)^n=\lambda^{n+1}\alpha\wedge(d\alpha)^n$
While reading to Geige's An Introduction to Contact Topology, I got stuck after the following statement:
Observe that $\alpha$ is a contact form precisely if $\alpha\wedge(d\alpha)^n$ is a volume ...
- 2,523
7
votes
0
answers
121
views
Combinatorial structure of the tesseract
On a cube, you can define the notion of "the next edge around this vertex, in clockwise order."
Formally, if $D$ is the space of darts (edges of the cube with one of the endpoint vertices ...
- 16.7k
0
votes
2
answers
133
views
Natural orientation of an underlying real vector space
Consider the following argument from a book (included as an image to show exactly what is printed):
There are a number of minor typos here, but I'm mainly concerned with the computation of $\Delta(...
- 2,274
1
vote
1
answer
38
views
orientation difference between two triangles in 3D space
Lets there be 2 sets of 3 points in 3D space, representing 2 congruent isosceles triangles. The apexes of both triangles are located at the point (0,0,0). How do I calculate the difference between the ...
3
votes
1
answer
99
views
Using Möbius band to define orientability of surfaces
Given a surface $S$, the usual definition of orientability is that there is an atlas of charts with all transition maps having non-negative Jacobian.
Donaldson, in his book on Riemann surfaces, says a ...
- 697
0
votes
0
answers
16
views
Hat Knot Floer Homology with Z coefficients calculation
I would like to ask if there is a reference which carries out the calculation of the hat knot Floer homology of a knot with $\mathbb{Z}$ coefficients, i.e., $\widehat{HFK}(K;\mathbb{Z})$, where the ...
2
votes
1
answer
64
views
Compatible oriented charts for all open balls in a manifold
Let $M$ be an orientable smooth manifold. Can I find for every set $U \subset M$ that is diffeomorphic to $\mathbb{R}^n$ a chart $\phi_U : U \to \mathbb{R}^n$ so that the transitions functions $(\...
- 1,498
0
votes
0
answers
37
views
Geometric intuition and interpretation of orientations in higher dimensions?
I've only come across orientations in the context of an analysis class talking about orientated triangulations of oriented affine k-simplex. We defined having a positive orientation as the determinant ...
0
votes
0
answers
22
views
Orientation-preserving order on vector space
Given a finite-dimensional real vector space V, is there a total order O on the vectors in V such that any basis of V, when ordered according to O, has the same orientation?
Does existence depend on ...
- 151
0
votes
0
answers
60
views
How to calculate pitch, yaw and roll using two sets of 3d points?
I am trying to calculate the pitch, yaw and roll of the forearm using two sets of two 3d(x,y,z) coordinates during movement, a set for the wrist and a set for the elbow. I would like to know how to ...
- 17
1
vote
0
answers
84
views
Every surface of revolution is orientable
I understand this proof to my question. But what if I have the following definition of "orientable manifold"?
Let $M$ be a manifold of class $C^k$, where $1\leq k \leq +\infty$. An atlas $\...
- 1,592
2
votes
1
answer
120
views
Orientation of a vector space from a homological point of view
I was reading Milnor and Stasheff's Characteristic Classes but I got stuck at the beginning of Chapter 9 when a homological interpretation of the orientation of a real vector space is given. More ...
- 866
9
votes
2
answers
294
views
In what sense is an orientation continuous?
I have been reading from Lee's Introduction to Smooth Manifolds and I have gotten to the section on how an orientation on a manifold is defined.
First Lee defines an orientation on a vector space $V$ ...
- 2,410
0
votes
0
answers
56
views
Under what condition is a coordinate chart orientation-preserving
Let $\phi: U \rightarrow O$ be a coordinate chart of an $n$-dimensional differentiable oriented manifold where $U$ is an open subset of the manifold and $O$ is an open subset of $\mathbb{R}^n$. If we ...
- 137