All Questions
Tagged with cotangent-bundles sg.symplectic-geometry
16 questions
41
votes
5
answers
5k
views
Can cotangent bundles see exotic smooth structures?
I have two questions that are inspired by a couple of questions here on MO (referenced below), as well as by a conversation with some other grad students at a summer school.
Caveat: I'm not a ...
- 10.9k
20
votes
1
answer
2k
views
Functoriality of the cotangent bundle
Recall that to any manifold $X$, I can assign in a canonical way a manifold $\mathrm T ^* X$, the total space of the cotangent bundle over $X$. Recall also that, unlike the tangent bundle ...
- 54.6k
11
votes
2
answers
4k
views
Cotangent bundle lift theorem
Let $M$ be a smooth manifold and $T^\ast M$ be its cotangent bundle. Consider the tautological 1-form $\theta$ on $T^\ast M$ ($\theta=\sum y_i dx^i$ in local canonical coordinate systems).
A ...
- 2,244
6
votes
2
answers
1k
views
symplectic structure of tangent bundle of $\mathbb{S}^{n-1}$
It is well known that $T\mathbb{S}^{n-1}$ is diffeomorphic to $M= f^{-1}(1)$ where
$f:\mathbb{C}^n\rightarrow \mathbb{C}$ is $f(z):=\sum_{i=1}^{n} z_{i}^{2}$.
Two questions:
1) Is $M$ a ...
- 366
6
votes
0
answers
368
views
Lagrangian up to Hamiltonian in cotangent bundle
I want to understand the folklore conjecture that, in a CY manifold, Lagrangians up to Hamiltonian isotopies are represented by special Lagrangians by examining cotangent bundle and Hodge theory.
...
- 61
5
votes
1
answer
994
views
Does the preimage of the Slodowy slice in $T^*G/P$ have a name?
Let $G$ be your favorite simple complex Lie group, and $P\subset G$ your favorite parabolic subgroup. We can identify $T^*G/P$ with the space of pairs $$\{(gP,x)\in G/P\times \mathfrak g | x\perp \...
- 44.7k
3
votes
1
answer
637
views
Can the class of the canonical bundle be recovered from the total space of the cotangent bundle if one forgets that it is a cotangent bundle?
This is a somewhat speculative question, so bear with that (or not, as is your preference).
Let $X$ be a smooth projective variety, and let $\omega_X$ be its canonical sheaf. The Euler class of ...
- 44.7k
3
votes
0
answers
99
views
Displacing a conormal Lagrangian from the zero section
I was told that the conormal bundle $\nu^*K$ of a knot $K\subset S^3$ can be displaced from the zero section $0_{S^3}$ in $T^*S^3.$ Having no intuition about whether/how often this happens in general, ...
- 1,687
3
votes
0
answers
394
views
Reference for plumbing cotangent bundles as Liouville manifolds
A Liouville domain $(W, \omega,\alpha, X)$ is a compact manifold $W$ with boundary $\partial W$, and a exact symplectic structure $\omega = d\alpha, \iota_X \omega = \alpha$, such that $X$ points ...
- 316
2
votes
0
answers
223
views
how to understand the manifold with boundary jet bundle and cotangent bundle with boundary
Suppose that $M\subset (W^{2n},\omega)$ is an $n$-dimensional manifold with smooth boundary $\partial M$, where $(W,\omega)$ is a $2n$-dimensional Kähler manifold and boundary with contact type ...
- 111
2
votes
0
answers
226
views
When is a symplectomorphism on a cotangent bundle the lift of a diffeomorphism on the base manifold [duplicate]
Let $X$ be a manifold and $T^\ast X$ the cotangent bundle. Let $\alpha$ denote the tautological $1$-form on $T^\ast X$ so that $(T^\ast X, \omega=-d\alpha)$ is a symplectic manifold. I want to know ...
- 243
2
votes
0
answers
643
views
A symplectic structure for cotangent bundle
Before that I mention my question explicitly, I start with my motivation:
Look at $\mathbb{D} \times \mathbb{C}=\{(x_{1},x_{2},y_{1},y_{2})\mid x_{1}^{2}+x_{2}^{2}< 1\}$.This can be identified ...
- 366
1
vote
1
answer
97
views
Derivative of the symplectomorphism evaluated at a point of the zero section of the cotangent bundle
It might be an easy question, possibly not worth posting here. In the proof of the Lagrangian neighborhood theorem, the authors have written the expression for the derivative of a symplectomorphism at ...
- 1,097
1
vote
0
answers
189
views
Relating the Morse index with the Maslov index
In the following paper https://arxiv.org/pdf/math/0408280.pdf there is created an isomorphism between the Floer Homology of an hamiltonian functional $H$ in the cotangent bundle and the the Morse ...
- 791
1
vote
0
answers
221
views
Co-normal bundle of orthogonal compliment
Is the following fact well known?
Let $X$ be a manifold and $V$ be a vector space. Let $E_1$ be a sub-bundle of the constant bundle $X \times V$. Let $E_2$ be its orthogonal compliment in $X \...
- 2,649
-4
votes
1
answer
468
views
Symplectic forms and 1-forms [closed]
Suppose we have a real symplectic manifold $(M,\omega)$. Under what conditions can we find a global 1-form $\alpha$ such that $\omega = \alpha \wedge\alpha$?
Obviously there are some simple ...
- 1,025