Questions tagged [orientation]

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

501 questions
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Orientability of $m\times n$ matrices with rank $r$

I know that $$M_{m,n,r} = \{ A \in {\rm Mat}(m \times n,\Bbb R) \mid {\rm rank}(A)= r\}$$is a submanifold of $\Bbb R^{mn}$ of codimension $(m-r)(n-r)$. For example, we have that $M_{2, 3, 1}$ is non-orientable, while some others are, such as…
26
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Does the Gauss-Bonnet theorem apply to non-orientable surfaces?

I found statements of the Gauss-Bonnet theorem here, here, here, here, here, here, here, and here. None of them require that the surface be orientable. However, Ted Shifrin claims in a comment to this question that the Gauss-Bonnet theorem actually…
19
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Can the interior of a manifold be orientable but not its boundary?

Suppose $M^m$ is a manifold with boundary. If we are given an orientation for $M$, we can then derive an orientation for $\partial M$ by considering the orientation of $TM$ at $\partial M$ and then using an outward-pointing vector to get an…
Eric Auld
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Understanding the orientable double cover

Definition: if $M$ is a smooth manifold, define the orientable double cover of $M$ by: $$\widetilde{M}:=\{(p, o_p)\mid p\in M, o_p\in\{\text{orientations on }T_pM\}\}$$ together with the function $\pi:\widetilde{M}\to M$ with…
rmdmc89
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12
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Meaning of the expression "orientation preserving" homeomorphism

The only time that I've heard the term "orientation-preserving map" was in Linear Algebra, but today I read the term orientation-preserving homeomorphism of the circle in the following context: If a homeomorphism $f$ of $S^1$ has a rational rotation…
u1571372
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Extrinsic and intrinsic Euler angles to rotation matrix and back

currently I'm working on the visualization of coordinate systems in space to understand rotation matrices better. Until now I thought everything would be ok, but there is a thing that does not get into my head, maybe you can help. I'll be as precise…
10
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3 answers

Fixed points in mapping from Möbius strip to disk [Explanation or reference needed]

One of the most elegant demonstrations in topology is the proof of the inscribed rectangle problem (a solved variant of the unsolved inscribed square problem) which states that for any plain, closed continuous closed curve $\Gamma$ in…
10
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Homology and cohomology of 7-manifold

I have the following problem: Let $M$ be a connected closed $7$-manifold such that $H_1(M,\mathbb{Z}) = 0$, $H_2(M,\mathbb{Z}) = \mathbb{Z}$, $H_3(M,\mathbb{Z}) = \mathbb{Z}/2$. Compute $H_i(M,\mathbb{Z})$ and $H^i(M,\mathbb{Z})$ for all $i$. I…
10
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Is there a orientable surface that is topologically isomorphic to a nonorientable one?

Is there a surface that is orientable which is topologically homeomorphic to a nonorientable one, or is orientability a topological invariant.
PyRulez
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Fundamental class of the connected sum of two closed orientable manifolds

I need to find a representation of $ [M \mathbin\sharp N] \in H_n(M \mathbin\sharp N) $ in terms of the fundamental classes $[M]$ and $[N]$. My idea is that $$ [M \mathbin\sharp N] = [M]+[N]$$ which is based on this MO question, whose answer is a…
Riccardo
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9
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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 a positive parametrization $\boldsymbol…
Ders
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A topological group which is also a (not necessarily smooth) manifold is orientable

I am trying to show that a topological group which is also a (not necessarily smooth) manifold is automatically orientable. I know of a proof involving transition functions for smooth manifolds, in which case the object in question is a Lie group. I…
9
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1 answer

The Antipodal Map is Orientation Preserving iff $n$ is Odd

The following result is well-known. Theorem. The antipodal map on $S^n$ is orientation preserving if and only if $n$ is odd. Below I provide a proof in which there must be an error since I reach to a wrong conclusion. Can somebody please point out…
caffeinemachine
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9
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Topological idea of orientability of manifold

While reading Poincare Duality a new idea of orientability of manifold came in my mind.I dont know wheather this idea is new or not, or even true or false. My idea is following... A $n$-dim manifold $X$ is called non-orientable if there exists an…
9
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Proving that the quotient manifold is orientable if and only if the group action is orientation-preserving

I'm trying to solve the following exercise in Lee's book. Suppose M is a connected, oriented smooth manifold and Γ is a discrete group acting freely and properly on M. We say the action is orientation-preserving if for each γ ∈ Γ, the diffeomorphism…
adrija
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