# Questions tagged [orientation]

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

289 questions
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### Metric to describe the “distance” between and orientation and position pair.

I have two points and would like a metric that tells me how close those points are to each other. Each point is described by both a 3D position and a 3D orientation. I can determine the distance ...
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### Non-orientable 1-dimensional (non-hausdorff) manifold

Is there any nice example of a 1-dimensional non-hausdorff manifold that is not oriented? I have tried the line with two origins, but maybe something more exotic is needed?
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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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### 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 ...
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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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### 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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### Rotation with roll pitch and yaw in different coordinate system

Say I am given a point in an x1,y1,z1 coordinate system. I have a different coordinate system, x2,y2,z2 that has the same origin as the x1,y1,z1 system, but the axis are not aligned. I have roll, ...
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### Torque required to achieve a desired quaternion

I was hoping someone can either explain or direct me towards a source that can help me with the following problem (not for homework, more of a hobby). Given an object with a current quaternion $q_c$ ...
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### Diffeomorphism Between Surfaces Preserves Orientability

From Do Carmo (Exercise 2.6.2). Let $S_2$ be an orientable regular surface and $\varphi:S_1\rightarrow S_2$ be a local diffeomorphism at every $p\in S_1$. Prove $S_1$ is orientable. Up until this ...
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### understanding orientable manifolds

I'm reading Warner. "Foundations of Differentiable Manifolds and Lie Groups." p. 138. I don't get the statement in the definition of orientable manifolds. 4.1 Definitions $\;$ (the preface omitted) ...
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### Smooth isotopy preserves orientation

Let $N$ a $n$-dimensional connected manifold and let $h: N \rightarrow N$ a diffeomorphism such that $h$ is smoothly isotopic to the identity map $\text{id}_N : N \rightarrow N$. It's clair that the ...
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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 ...