Questions tagged [orientation]
For question regarding the notion of orientation both in topology and in global analysis.
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Why are quasiconformal maps orientation preserving under the analytic definition?
Quasiconformal maps admit a lot of equivalent definitions.
One of the geometric definitions states that a homeomorphism $f$ is quasiconformal, iff it preserves orientation and changes the modulie of ...
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$\Lambda^n(M)$ is not isomorphic to $C^{\infty}(M)$ if M is not orientable
Let $M$ be a differentiable Manifold of dimension n. If $M$ is orientable, then there exists an $\omega \in \Omega^n(M)$ (top-degree differential form) such that $\omega(p) \neq 0$ $ \forall p \in M$.
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What general concept, other than oriented volume, embodies all even (odd) permutations of $n$ elements?
Given an ordered orthogonal basis $v_1...v_k$ of a $k$-dimensional vector space, we can construct a $k$-vector in the geometric algebra over that space, simply by multiplying the basis vectors in the ...
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Orientation covering
My question is about whether or not the image of an open set is still an open set. I'm going to write the construction of the orientation covering and then I'll ask what I can't figure out.
Let $M$ be ...
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Equivalent definition for bundle orientability.
We know there are several equivalent definition for orientability of manifolds(see Lee's introduction to smooth manifolds):
Existence of a choice of orientation at each point $p\in M$, such that ...
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Pre image orientation and smooth map
I consider a smooth map $ F: X\to N$ where $X$ and $N$ are smooth, oriented manifolds of dimension $m+1$ and $m$ respectively, $X$ being compact and with boundary $\partial X=M$.
For a regular value $...
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Collapse of the AHSS for $E^*(\mathbb{C}\mathrm{P}^n)$ for $E$ a complex-oriented ring spectrum [duplicate]
Denote by $\newcommand{\CP}{\mathbb{C}\mathrm{P}}i_{n, m}\colon \CP^n \to \CP^m$, $n \leq m \leq \infty$ the canonical inclusion.
Let $E$ be a homotopy commutative ring spectrum and $t \in \tilde{E}^2(...
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Justification of the induced orientation on a sub-manifold with boundary
In my study of orientation of sub-manifold, I tried to construct the induced orientation on the boundary of a sub-manifold and I would like to have your advice because in the book I have (Milnor’s ...
- 1,381
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Left or right for 2 points on a line, clockwise or counterclockwise for 3 points in the plane. What is the analogue for 4 points in space?
For $a, b \in \mathbb{R},$ there is a notion of left or right. As a society, we agreed $a$ is left of $b$ if $a<b$ and otherwise $a$ is right of $b.$ It could've been the other way and nothing ...
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Understanding proof of orientability of the level set of a regular value in $\mathbb{R}^{n+1}$
Given $f: \mathbb{R}^{n+1} \to \mathbb{R}$ a $C^{\infty}$ function with 0 being a regular value, we know that $M = f^{-1}(0)$ is a manifold of dimension $n$. There is an answer given here, though I ...
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Lee's definition of a consistently oriented basis for a real vector space $V$ of dimension $n\geq 1$
Let $V$ be a real vector space of dimension $n\geq 1$. Lee defines in his book Introduction to Smooth Manifolds chapter 15 that
We say that two ordered bases $(E_1,\dots,E_n), (\tilde{E}_1,\dots,\...
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Curve is traveled clockwise or anti-clockwise
Given the curve
$$
\vec{\mathbf{r}}(t)
= \bigl(t - \tfrac{1}{3}t^3 \bigr) \, \vec{\mathbf{i}}
+ (4 - t^2) \, \vec{\mathbf{j}},
$$
how can I tell whether it's traveled clockwise or counterclockwise?
...
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Any manifold with fundamental group isomorphic to $\mathbb{Q}$ is orientable
I have enjoyed the construction of the rational circle found here, but I still have no clue on how to show that such a space, or any other with fundamental group $\mathbb{Q}$, must be orientable.
In ...
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Any smooth compact 2-dimensional submanifold $S\subset\mathbb{R}^3$ is orientable
A manifold $M$ is orientable if the bundle of antisymmetric $n$-linear forms $\Lambda^n(M)$ is trivial. Equivalently, $\Lambda^n(M)$ admits a nowhere vanishing section.
I suppose we must assume that $...
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Understanding the Representation of Signal Variation by $\hat{u}^T T \hat{u}$ in the Context of 3D Orientation Tensors
I was studying the Gunnar Farneback method for dense optical flow and encountered the following paragraph:
By stacking the frames of an image sequence onto each
other we obtain a spatiotemporal image ...
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