23
votes
Examples of interesting non orientable closed 3-manifolds
One gets non-orientable closed 3-manifolds by taking a non-orientable surface, and crossing with $S^1$, such as $P^2\times S^1$.
In fact, the geometrization theorem hasn't been proven completely for ...
17
votes
Accepted
Does the Gauss-Bonnet theorem apply to non-orientable surfaces?
The answer is already given in the comments (by Ryan Budney and Mizar). But I think it makes sense to clear this confusing point. The classical Gauss-Bonnet formula is [e.g. https://en.wikipedia....
15
votes
Accepted
Does an oriented $S^3$ fiber bundle admit the structure of a principal $SU(2)$-bundle?
No. (The main idea here is present in Dylan Wilson's comment.)
Every principal $SU(2)$-bundle over $S^2$ is trivial, because $\pi_1 SU(2)$ is trivial. But there is a nontrivial oriented bundle over $...
11
votes
Accepted
Reference on complex cobordism
This is worked out in part 2 of
Adams, J. F., Stable homotopy and generalised homology, Chicago Lectures in Mathematics. Chicago - London: The University of Chicago Press. X, 373 p. £ 3.00 (1974). ...
11
votes
Accepted
A topological group which is also a (not necessarily smooth) manifold is orientable
In fact beside your question there is a beautiful theorem in homotopy theory. This theorem due to T. Bauer, N. Kitchloo, D. Notbohm and E. K. Pedersen guarantees that any loop space $X=\Omega B$ where ...
11
votes
Examples of interesting non orientable closed 3-manifolds
A curiosity is obtained as follows: take a solid cube $[-1,1]^3$ and identify one pair of two opposite faces by a symmetry with respect to a coordinate axis, while identifying the other two pairs of ...
8
votes
Accepted
Is there a well-known notion of orientability for finite geometries?
To do that you would need a notion of a non-orientable and an orientable linear transformation, i.e., essentially a notion of a "positive" and "negative" determinant, where "positive" determinants ...
4
votes
On the “Non-conservation of parity in weak interactions”
I don't think the mathematics needs to be more involved than to appreciate the difference between an axial vector (or pseudovector) and a polar vector. An axial vector does not change sign upon ...
3
votes
Precise mathematical relation between chirality (or $\gamma_5$) and (spatial) orientation in $1+3$ Minkowski spacetime
Chirality is related to orientation via spin-momentum locking.
The projector $P_\pm=\tfrac{1}{2}(1\pm\gamma_5)$ projects a spinor onto two subspaces that are decoupled in the evolution equation of a ...
3
votes
Accepted
Orientation bundle and its flat connection
There is a different construction of orientation bundles. One considers the $\{\pm1\}$-principal bundle $o(TM)$ of fibrewise orientations of $TM$. The associated real line bundle $o(TM)\times_{\pm 1}\...
3
votes
An orientable surface that cannot be embedded into $\Bbb R^3$?
It seems to me that every orientable surface is indeed embeddable in $\mathbb{R}^3$. By Ian Richards' classification theorem (https://www.ams.org/journals/tran/1963-106-02/S0002-9947-1963-0143186-0/...
3
votes
Accepted
Associativity of orientations of determinant bundles in Floer homology
The issue here are the isomorphisms in your suggested proof. There are choices and conventions involved. To give names to the isomorphisms clarifies what needs to be checked.
$$
\iota_{K,L}:Det(K\...
3
votes
Accepted
3
votes
Orientability of orbit type strata of Lie group actions
In your question, you may as well take $M=M_j=$ a ($G$-invariant) tubular neighborhood of a single orbit $G/S$. Near the point $S/S$, the space looks like $\mathfrak g/\mathfrak s \times N$, where $N$ ...
2
votes
Accepted
non-orientability of vector bundles induced from a symmetric group action
Yes if $M$ is connected, no otherwise.
If $M$ is connected, the covering gives a surjective homomorphism $\pi_1(M/\Sigma_k)\to \Sigma_k$. The vector bundle comes from a representation of $\pi_1$ ...
2
votes
Is there a notion of a connection for which the horizontal lift of a curve depends on its orientation?
If you look at the Ehresmann connection in T(TM) (or T(TM\0)) associated with a second order differential equation or a Finsler metric (which is not necessarily reversible), then this is orientation ...
2
votes
Finding a volume form on a fibre of a submersion between oriented manifolds
Take a unit tangent p-vector $a$ based at $y \in Y$ (you need the volume form on $Y$ to say what a unit p-vector is). At each point $x$ of the fiber $f^{-1}(y)$ consider the contraction of your volume ...
1
vote
Exponential trigonometric integral
Here are my comments as an answer: Based on my short study as well as also on the comment by @TheSimpliFire, I am quite sure that the best one can get is the integral representation below, or a ...
1
vote
Accepted
Necessary and sufficient conditions for pseudo Riemannian manifold to be time orientable
For the first question I believe the answer is yes. Almost certainly it's in the literature but I do not know this literature very well.
The idea is as you suggested. Maximal-rank timelike (or ...
1
vote
On the “Non-conservation of parity in weak interactions”
If some interaction acts differently on the right-handed and left-handed components of a particle, the interaction is automatically parity violating. So you just need to look at the Lagrangian and ...
1
vote
Orientability and higher dimensional Moebius strip
Yes. If all loops give rotations as holonomy, then the holonomy group lies in the rotation group, so the manifold is orientable.
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