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Questions tagged [orientation]

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

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A “parallel manifold” is always orientable

I want to solve the following problem from Spivak's Calculus on Manifolds: Let $M$ be an $(n-1)$ dimensional manifold in $\mathbb{R}^n$. Let $M(\varepsilon)$ be the set of end points of normal ...
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2answers
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Is orientability needed to define volumes on riemannian manifolds?

In the book Riemannian Geometry, by Mandredo do Carmo, he supposes that $M$ is a riemannian oriented manifold and then defines the volume of a region $R$ contained in some image $\boldsymbol x(U)$ of ...
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What does the 2nd degree derivative of a cubic Bezier curve actually represent?

I have a $3D$ Bezier curve. Each co-ordinate along its path is defined by the equation: $$ f(t) = t^3 \bigl(a_2+3(c_1-c_2)-a_1\bigr) + 3t^2 (a_1-2c_1+c_2) + 3t(c_1-a_1) + a_1 $$ where $a_1, a_2$ are ...
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1answer
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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 ...
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176 views

Is “Let $M$ is a connected n-manifold, then: $M$ is $R$-orientable iff $H_n(M;R)\cong R$” true?

I heard that every topological $n$-manifold $M$ is $\mathbb{F}_2$-orientable, but then for $M=\mathbb{R}^2$ is must be $H_2(\mathbb{R}^2;\mathbb{F}_2)\neq 0$? In lecture we had the lemma: Let $M$ is ...
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1answer
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Is the total space of the tautological line bundle over $\mathbb{R}P^{n}$ a non orientable manifold?

Is it true to say that the total space of the tautological line bundle over $\mathbb{R}P^{n}$ is a non orientable manifold? Perhaps the question can be indirectly related to the following question:...
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Fundamental group of $M$ has no subgroup of index $2\Rightarrow M$ is orientable

Let $M$ be a connected smooth manifold such that, for every $p\in M$, the fundamental group $\pi_1(M,p)$ has no subgroup of index $2$. Prove that $M$ is orientable. Here's what I know: there is a ...
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Integrate form $\omega=x_jdx_{1}\wedge\cdots\wedge dx_{i-1}\wedge dx_{i+1}\wedge\cdots dx_n$ on $S=\{\textbf{x}\in\mathbb{R^n}:||\textbf{x}||=1\}$

Integrate form $\omega=x_jdx_{1}\wedge\cdots\wedge dx_{i-1}\wedge dx_{i+1}\wedge\cdots dx_n$ on $S=\{\textbf{x}\in\mathbb{R^n}:||\textbf{x}||=1\}$. $i,j$ are constant. Doing it from definition doesn'...
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1answer
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Orientability of the total space of a vector bundle and total space of its sphere bundle

Let $\xi \colon E \to B$ be a (finite dim.) vector bundle and let $\pi \colon S(E)\to B$ the restriction to its sphere bundle. In particular, if $i\colon S(E)\to E$ is the embedding, we have $\xi \...
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1answer
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Orientable manifold using definition.

I'm reading the Do Carmo book in the section of the orientable surfaces, but I still don't understand the idea of orientability, because the examples he uses to clarify the concept doesn't use ...
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1answer
525 views

Oriented atlas on a circle

I'm trying to find an oriented atlas on the circle $S^1$, i.e., I want to find an atlas for $S^1$ such that for any two overlapping charts $(U,s)$ and $(V,t)$ of the atlas, the derivative $d s/d t$ ...
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1answer
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Orientability of triangulated manifold

Suppose $M$ is a topological manifold, triangulated in the sense of CW complexes. If $M$ is closed and connected, I am able to prove (by using the cellular homology and its boundary map) that ...
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1answer
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Typo in Lee's Introduction to Smooth Manifolds, Exercise 15.30?

Exercise 15.30 (p.389): Suppose (M, g) and ($\tilde{M}, \tilde{g}$) are positive-dimensional Riemannian manifolds with or without boundary, and $ F: M \to \tilde{M}$ is a local isometry. Show that $ ...
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1answer
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Let $p=(5,0,-4)$ and $v \in T_{(5,0,-4)}M$. Compute $(F^{*}\omega)_p(v)$.

Let me show my work before presenting the problem itself. Let $M=\{(x,y,z) \in \mathbb{R}^3 : x+y=5, x+z=cos^2y\}$. We can easily see that $M$ is a submanifold of $\mathbb{R}^3$ of dimension $1$. ...
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The boundary orientation coincides with the preimage orientation

Prove that the boundary orientation of $S^k=\partial B^{k+1}$ is the same as its preimage orientation under the map $g: \mathbb R^{k+1}\to \mathbb R$, $g(x)=|x|^2$. The boundary orientation is ...
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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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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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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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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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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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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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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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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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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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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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When is the orientation presheaf a sheaf?

The orientation presheaf of a topological $n$-manifold $X$ is $$U \mapsto H_n(X, X-U)$$ The manifold $X$ is orientable iff there exists a global section which is a generator of each stalk (I believe)...
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1answer
59 views

Show that $\mathbb{S}^{n+m}$ is not homeomorphic to a product of orientable manifolds

I want to prove that the sphere $\mathbb{S}^{n+m}$ is not homeomorphic to the product of N and M, orientable manifolfs with $\textit{dim}\;N=n$ and $\textit{dim}\;M=m$. I know that I have to use the ...
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2answers
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What does it mean for a (non-smooth) homeomorphism between oriented smooth manifolds to be orientation preserving?

The definitions I know of orientability of manifolds are in terms of tangent spaces. However, for example in this answer there is mention of orientation preserving homeomophisms (between orientable ...
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1answer
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extended kalman filter equation for orientation quaternion

I have a body pose data sampled with a given frequency. Using model with constant velocity motion between frames i filter position with EKF. State equation is given by: $$ \begin{pmatrix} x_{k+1} \\ ...
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1answer
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Are closed (topological) submanifold in $\mathbb R^n$ of codimension 1 orientable?

See Are closed, properly embedded manifolds of co-dimension 1 in $\mathbb{R}^n$ orientable? for treatment of the smooth case. If the topological case of Jordan theorem holds for such manifolds, ...
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Orientability in N Dimensions

Let's say I have a set of $N$ points which define an ($N-1$) dimensional triangle (or N dimensional surface if you wish). I define a normal to that surface pointing in a specific direction. I want to ...
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1answer
63 views

Proof of : “Signature of $\mathbb{C}P^{2n}$ is $1$”

I started learning about signature of a $4k$-manifold and one of the most common example is the signature of $\mathbb{C}P^{2n}$. The only reference I found is tom Dieck's Algebraic Topology. Even ...
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1answer
63 views

Orientability of an n dimensional manifold

This question looks like pretty easy and missing a crucial point to solve it but I am not sure. Here it is: Let $M$ be an n dimensional compact manifold such that $H_i(M;\mathbb{Z}_2) = H_i(S^n;\...
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1answer
186 views

Quaternions disadvantages in Quadrotor UAV control on $SE(3)$

I am reading a paper which deals with the Geometric Control on $SE(3)$ of a Quadrotor UAV. At some point it says: Quaternions do not have singularities but, as the three-sphere double-covers the ...
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1answer
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How to prove that $MU$ is an oriented spectrum? A doubt in the proof in Kochman's book

I want to show that the Thom spectrum $MU$ is oriented, namely I want to find a class $x \in \widetilde{MU}^2(\mathbb{C}P^{\infty})$ whose restriction to $\widetilde{MU}^2(S^2)$ is a generator. in his ...
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1answer
53 views

Orientability of $\gamma^n\oplus \gamma^n$ WITHOUT characteristic classes

I was curious to find an argument to show orientability of the $2n$-bundle $$\gamma^n\oplus \gamma^n$$ where $\gamma^n$ is the canonical $n$-bundle over the infinite grassmannians $Gr_n(\mathbb{R}^{\...
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1answer
113 views

Sufficient condition for $\mathbb{Z}$-orientability

Let $X$ be a topological $n$-manifold. Let's define a R-orientation on $X$ as a choice of generators $\alpha_{x}\in H_{n}(X,X\setminus\lbrace x\rbrace;R)$ that is consistent. Suppose that $X$ is $\...
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1answer
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Question of deck transformation on double cover $\tilde{M}$ of non-orientable manifold $M$.

Suppose $(M,g)$ is a non-orientable, compact, connected Riemannian manifold with positive sectional curvature, $\tilde{M}$ is its orientable double cover. $\varphi$ is deck transformation of $\tilde{M}...
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1answer
33 views

$df_x$ maps $N(S.X)$ isomorphically

This is actually just a linear algebra problem, but this text is taken from Guillemin and Pollack, Differential Topology on page 100. Real Problem: Let $f: X \to Y$ be smooth, $S = f^{-1}(Z)$ where ...
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1answer
164 views

How do I determine whether the orientation of a basis is positive or negative using the cross product

I know that if I have an orthonormal base in $\mathbb{R}^3$, namely $e_1$, $e_2$ & $e_3$, then it is positively oriented if $$e_1 \times e_2 = e_3$$ $$e_2 \times e_3 = e_1$$ $$e_3 \times e_1 = ...
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1answer
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Let $x$ and $y$ are linearly interdependent and that $x+y+z = 0$. Show that $\theta(x,y) + \theta(y,z) + \theta(z,x) = 2\pi$

In the book of Linear Algebra by Werner Greub, at page 202 Q.11.a, it is asked that Let $x,y,z$ be three vectors of a plane such that $x$ and $y$ are linearly interdependent and that $x+y+z = 0$. ...
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1answer
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Construct an volume form on $\mathbb RP^n$ (n odd) using the antipodal map

We have the volume form $\mu=i^*(\omega)$ on $S^n$, where $$\omega=\sum_{j=1}^{n}(-1)^{j-1}x_{j}dx_{1}\wedge\cdots\wedge \hat{dx_{j}}\wedge\cdots dx_{n+1}$$ is a k-form on $\mathbb R^{n+1}$ and $i:S^...
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1answer
214 views

Moebius strip and orientability

Lemma: Let $M=U\cup V$ be a smooth manifold with $U$ and $V$ open, connected, oriented and such that $U\cap V$ has two connected components $W_1$ and $W_2$. Then $M$ is orientable $\Leftrightarrow$ ...
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1answer
206 views

An orientation on a $(n-1)$-dimensional submanifold.

This question goes on where this question ended. Given is an non-empty $(n-1)$-dimensional ($n\ge2$) differentiable submanifold $X\subset\mathbb{R}^n$ such that there exists an open $U\subset\mathbb{R}...
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1answer
448 views

calculating the orientation of an object

If you have a rotation matrix (or an attitude/direct cosine matrix, which are all synonyms). This matrix actually transforms vectors from one reference frame to another. But if your goal is to know/...
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1answer
520 views

First homology group of non-orientable manifold

Can I use Poincaré duality to prove that the first homology group of a non-orientable manifold $M$ is not zero? I only need to prove that $H_1(M;\mathbb Z_2)\ne0$, and by the universal coefficient ...
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1answer
65 views

Can the twisting of mobius band be represented by a U (1) bundle?

With the usual embedding of a mobius band, the strip is twisted by an angle pi, smoothly, as it goes round.I think this can be represented intrinsically, independent of the embedding, by attaching a ...
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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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0answers
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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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1answer
58 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 ...