# Questions tagged [orientation]

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

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### Proof construction for non-orientability of Mobius strip

I'm following Spivak's comprehensive guide to differential geometry and I'm getting a bit stuck on a calculation about orientability. The example is showing that the Mobius strip considered as a line ...
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### need a rigorous proof of the generator of local homology group $H_{n}(\mathbb{R}^n,\mathbb{R}^n-\{0\})$ is the given simplex

My question: Why the generator of local homology group $H_{n}(\mathbb{R}^n,\mathbb{R}^n-\{0\})$ is that simplex $\sigma$, where $\sigma=[v_0,e_1,\cdots,e_n]$, $v_0=(-1,-1,\cdots,-1)\in \mathbb{R}^{n}$ ...
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### What are oriented embeddings of zero-dimensional manifolds?

I'm trying to understand what it should mean for a submanifold of an oriented manifold to have a compatible orientation. For $1$-dimensional manifolds embedding into a $2$-dimensional manifold, I ...
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### Orientation reversing diffeomorphism but no isometry?

Is it possible that an oriented Riemannian manifold $(M,g)$ with a large isometry group $\text{Isom}(M,g)$ has an orientation reversing self-diffeomorphism but no orientation reversing self-isometry, ...
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### Consistency between two different ways of defining an orientation on $S^n$

In the book Topology, Geometry and Gauge Fields Interactions by Gregory L. Naber, he defines the standard orientation of $S^n$ (viewed as a subset of $\mathbb{R}^{n+1}$), $n\geq 2$ by the oriented ...
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### Non-vanishing of mod 2 intersection number for odd-dimensional manifolds implies the ambient manifold is nonorientable.

Let $X \subset Y$ be a compact embedded manifold of odd dimension such that $\dim X = \frac12 \dim Y = n$. At the end of the section titled "Oriented Intersection Theory", Guilleman and ...
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### Orientation preserving isometries of the sphere

I know that the isometries of $\mathbb{S}^n\subseteq\mathbb{E}^{n+1}$ are $O(n+1)$. Intuitively, I think that $SO(n+1)$ should be the orientation-preserving ones. But I'd like to prove this rigourosly....
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### Induced orientation given a short exact sequence (in Morse homology over $\mathbb{Z}$)

I am new to the field of algebraic topology and am currently studying Morse homology for a project. I read that given a short exact sequence $$0 \to A \to B \to C \to 0$$ if we know the orientation ...
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### Orientation preserving parametrisation of torus

EDIT: After having posted the question along with an attempt at a solution, I could spot the mistake in my solution (it's easier to read LaTeX than your own imperfect handwriting...), so now it is ...
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### 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 ...
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### 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 ...
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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 $\omega_p(v_1, \... • 43 2 votes 1 answer 89 views ### Is there a natural way to build a volume form on an oriented Poisson manifold? Editet question Let$(M, \pi)$be an oriented Poisson manifold. Is there a natural way to build a volume form from the Poisson bivector$\pi$? Original question It is always possible to build a volume ... • 493 1 vote 0 answers 107 views ### Show that$\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{... • 8,651 0 votes 0 answers 58 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: ... • 95 0 votes 1 answer 48 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 146 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 ... • 2,264 0 votes 0 answers 49 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 38 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,... • 105 0 votes 1 answer 147 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 ... • 103 0 votes 0 answers 136 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 228 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 ... • 4,214 0 votes 0 answers 44 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 57 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 142 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 ... • 5,627 0 votes 1 answer 77 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$... • 57 0 votes 0 answers 133 views ### In proof of the Stokes's Theorem for Chains ( John Lee's Smooth manifolds ) I am reading the John Lee's Introduction to smooth manifold, second edition, Theorem 18.12 (Stokes's Theorem for Chains) and stuck at understanding some argument : I'm struggling with hours to ... • 2,264 3 votes 0 answers 48 views ### Group of Oriented Edges of a Tiling This is an idea I'm sure exists already, but is quite complicated, so it's hard to find without appropriate terminology. We consider a tiling of the plane by regular$n$-gons, containing an edge$e_0$... • 2,326 1 vote 1 answer 137 views ### Characteristic Property of the Orientation Covering (Problem 15-10 in Lee's Smooth Manifolds book) The problem statement: Let$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$is any ... • 4,445 1 vote 0 answers 33 views ### Effect on transition functions when changing the orientation of a manifold Let$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}...
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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 ...