# Questions tagged [orientation]

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

289 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-...
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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 ...
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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 ...
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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 ...
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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 ...
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### Partition of unity and volume form on a manifold

A smooth manifold $M$ is orientable iff there exists a nowhere-vanishing top form (i.e. volume form). In a coordinate chart $U\subset M$ we can find a volume form over $U$ that corresponds to the ...
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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 ...
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### 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 ...
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### A space that deformation retracts into the cylinder and Möbius band doesn't embed in $\Bbb R^3$.

Consider the Möbius band, and take the middle circle in it (so that it deformation retracts onto it). Glue the upper boundary of a cylinder through it. This gives a space that deformation retracts ...
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### A correspondence between generators of $H_n(\mathbb{R}^n,\mathbb{R}^n-\{0\})$ and eq. classes of orthonormal frames

The problem is about (topological) orientation of $\mathbb{R}^n$: Define an equivalence relation on orthonormal frames in $\mathbb{R}^n$ by declaring two frames equivalent if the matrix expressing ...
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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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### Confused (disoriented?) by questions about orientation

I believe I have a reasonable basic understanding of orientation. Yet, i'm finding myself utterly confused when facing a specific question. Here are several examples: Exhibit an ordered basis of ...
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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.
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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 ...
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### What would be the concept of “Infinite Dimension Orientability”?

We have the following definition of orientability for finite dimensional vector spaces: Definition: Let $\Bbb{E}$ be a finite dimension vector space and $\mathscr{B}(\Bbb{E})$ be the set of all ...
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### $M \times N$ orientable iff both $M$ and $N$ are orientable proof in terms of volume forms [duplicate]

I'm studying differential forms, and in my homework I'm asked to show that the product of two manifolds $M \times N$ is orientable if and only if both $M$ and $N$ are orientable. I want to show this ...
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### If $\omega \in \Omega^{n-1}(M)$ then $d_p\omega = 0$ at some $p \in M$.

Let $M$ be a compact, oriented, smooth manifold of dimension $n$. I have to show that if $\omega \in \Omega^{n-1}(M)$ then $d_p\omega = 0$ at some $p \in M$. My first attempt was to use the ...
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### A vector bundle which has an orientation-reversing isomorphism has a subbundle of rank $1$?

Let $E$ be a smooth real vector bundle of even rank, over a smooth manifold $M$. Suppose there exist an orientation-reversing vector bundle isomorphism $\Phi:E \to E$. Is it true that $E$ has a ...
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### Every oriented vector bundle admits an orientation reversing isomorphism?

Let $E$ be a real oriented vector bundle over a smooth manifold $M$. Is there a vector bundle isomorphism $\Phi:E \to E$ which reverses the orientation? (I know there are oriented manifolds with no ...
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### Immersion of non-orientable manifold in a small orientable one

I was trying to prove the following fact: given a non orientable manifold $M$ of dimension $m$, $M$ is always contained in an orientable manifold of dimension $m+1$. I have gotten nothing out of it, ...
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### How does one orient a simplicial complex?

I have a simplicial complex, built out of hyper-tetrahedra (5-cells) with the topology of $S_{4}$ and I would like to assign an ordering to it's vertices (some couple thousand), so that I can apply a ...
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### Direct sum of non-orientable bundles is orientable?

Let $M$ be a smooth manifold, and let $E_1,E_2$ be two non-orientable vector bundles over $M$. Is $E_1 \oplus E_2$ orientable? I am sure there is an easy answer, but somehow my search didn't result ...
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### How to define orientation on infinite dimensional vector space

Let $\mathbb{V}$ be a real Banach space (if someone knows the answer for more arbitrary T.V.S. then great). Is there some concept of orientation that one could define on $\mathbb{V}$ that matches the ...
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### Orientability of variety

Consider the real projective variety on $\mathbb{R}P^n \times \mathbb{R}P^n$ defined by the bihomogeneous polynomial $x_0y_0+...+x_ny_n$, i.e. $V=\{([x_0:...:x_n],[y_0:...:y_n])|x_0y_0+...+x_ny_n=0\}$....
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### $\Phi:M\to M$ be a smooth map whose restriction to $\partial M$ is the identity

Let $M$ be a connected oriented smooth n-dimensional manifold-with-boundary, and let $\Phi:M\to M$ be a smooth map whose restriction to $\partial M$ is the identity. If $\omega$ is any smooth n-...
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### Induced orientation on manifold

Suppose $M$ is a naturally oriented $n$-manifold embedded in $\mathbb{R}^n$ with non-empty boundary. Give $\partial M$ the induced orientation. The induced orientation defines a unit normal vector ...
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### Stokes' theorem: Induced orientation on the boundary of a manifold

The Question Let $K = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \geq 1\}$, where $K$ is oriented via the canonical volume form on $\mathbb{R}^3$: $dx \wedge dy \wedge dz$. Let $\mathbb{S}^2$ be ...
### Every smooth map $S^4 \to\mathbb{CP}^2$ has degree zero
I'm trying to show that every smooth map $f: S^4 \to \mathbb{CP}^2$ has degree zero. I'm not sure how to go about this, but I know that the oriented intersection number between any two closed 2-...