Questions tagged [orientation]

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

295 questions
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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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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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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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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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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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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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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 $d^{i}\: : \... 0answers 103 views Continuity of point wise orientation Prove that$[(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$, the ... 3answers 319 views When is the orientation presheaf a sheaf? The orientation presheaf of a topological$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)... 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 ... 2answers 78 views 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 ... 1answer 1k views 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: $$\begin{pmatrix} x_{k+1} \\ ... 1answer 189 views Are closed (topological) submanifold in \mathbb R^n of codimension 1 orientable? See Are closed, properly embedded manifolds of co-dimension 1 in \mathbb{R}^n orientable? for treatment of the smooth case. If the topological case of Jordan theorem holds for such manifolds, ... 2answers 54 views Orientability in N Dimensions Let's say I have a set of N points which define an (N-1) dimensional triangle (or N dimensional surface if you wish). I define a normal to that surface pointing in a specific direction. I want to ... 1answer 63 views Proof of : “Signature of \mathbb{C}P^{2n} is 1” I started learning about signature of a 4k-manifold and one of the most common example is the signature of \mathbb{C}P^{2n}. The only reference I found is tom Dieck's Algebraic Topology. Even ... 1answer 63 views Orientability of an n dimensional manifold This question looks like pretty easy and missing a crucial point to solve it but I am not sure. Here it is: Let M be an n dimensional compact manifold such that H_i(M;\mathbb{Z}_2) = H_i(S^n;\... 1answer 186 views Quaternions disadvantages in Quadrotor UAV control on SE(3) I am reading a paper which deals with the Geometric Control on SE(3) of a Quadrotor UAV. At some point it says: Quaternions do not have singularities but, as the three-sphere double-covers the ... 1answer 48 views How to prove that MU is an oriented spectrum? A doubt in the proof in Kochman's book I want to show that the Thom spectrum MU is oriented, namely I want to find a class x \in \widetilde{MU}^2(\mathbb{C}P^{\infty}) whose restriction to \widetilde{MU}^2(S^2) is a generator. in his ... 1answer 53 views Orientability of \gamma^n\oplus \gamma^n WITHOUT characteristic classes I was curious to find an argument to show orientability of the 2n-bundle$$\gamma^n\oplus \gamma^n$$where \gamma^n is the canonical n-bundle over the infinite grassmannians Gr_n(\mathbb{R}^{\... 1answer 113 views Sufficient condition for \mathbb{Z}-orientability Let X be a topological n-manifold. Let's define a R-orientation on X as a choice of generators \alpha_{x}\in H_{n}(X,X\setminus\lbrace x\rbrace;R) that is consistent. Suppose that X is \... 1answer 92 views Question of deck transformation on double cover \tilde{M} of non-orientable manifold M. Suppose (M,g) is a non-orientable, compact, connected Riemannian manifold with positive sectional curvature, \tilde{M} is its orientable double cover. \varphi is deck transformation of \tilde{M}... 1answer 33 views df_x maps N(S.X) isomorphically This is actually just a linear algebra problem, but this text is taken from Guillemin and Pollack, Differential Topology on page 100. Real Problem: Let f: X \to Y be smooth, S = f^{-1}(Z) where ... 1answer 164 views How do I determine whether the orientation of a basis is positive or negative using the cross product I know that if I have an orthonormal base in \mathbb{R}^3, namely e_1, e_2 & e_3, then it is positively oriented if$$e_1 \times e_2 = e_3e_2 \times e_3 = e_1e_3 \times e_1 = ... 1answer 61 views 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$x+y+z = 0$. ... 1answer 155 views 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^...
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 $\Leftrightarrow$ ...