3
votes
1answer
58 views

Looking for a good book on Morse-Bott functions.

I am looking for a book to study for the first time Morse-Bott functions. Does anyone know one that is easy to follow and detailed? If there is one connecting this subject with symplectic geometry, it ...
0
votes
0answers
25 views

Global section $$s:T^*G\to Sp(T^*G)$$

Let $G$ be a Lie group and then how can we define the global section $$s:T^*G\to Sp(T^*G)$$ Where $Sp(T^*G)$ means symplectic frame bundle of $T^*G$.
1
vote
1answer
62 views

Symplectic submanifolds in $\mathbb{R}^{4}$

Which symplectic submanifolds can be realized in $\mathbb{R}^{4}$? It easy to show that such submanifolds aren't compact. So, they are spheres with some handles and holes. Which relations between the ...
1
vote
2answers
346 views

Regarding Legendre transform from tangent bundle to cotangent bundle

(I'm a complete beginner at differential geometry) I'm studying about constrained systems, in which we "map a Lagrangian system from a tangent to a cotangent bundle. Hamiltonian dynamics then appears ...
3
votes
2answers
105 views

Proving that a particular submanifold of the cotangent space is Lagrangian

I have the following problem in my differential topology class: Let $M$ be an $n$-dimensional manifold and let $\omega$ denote the standard symplectic form on the cotangent space $T^*M$. Let $f \in ...
1
vote
0answers
130 views

Energy displacement of a cylinder is at most $\pi r^2$.

I want to show that the energy displacement of $Z^{2n}(r)$ is at most $\pi r^2$. In the textbook of Mcduff and Salamon they write that I should identify the two dimensional ball (with radius $r$) ...
5
votes
1answer
280 views

$4$-form $ \omega \wedge \omega$ vanishes on $S^4$

If $\omega$ is a closed $2$-form on $S^4$, how can I show the $4$-form $ \omega \wedge \omega$ vanishes somewhere on $S^4$? I am guessing that the fact we're talking about the $2$-form being ...
10
votes
2answers
334 views

Showing that some symplectomorphism isn't Hamiltonian

I have the next symplectomorphism $(x,\xi)\mapsto (x,\xi+1)$ of $T^* S^1$, and I am asked if it's Hamiltonian symplectomorphism, i believe that it's not, though I am not sure how to show it. I know ...
0
votes
0answers
72 views

Maslov Index product property.

I am not sure how to show the following property of the Maslov Index, which in McDuff and Salamon's Introduction to Symplectic Topology, Theorem 2.35, is called the product property. Let $\Lambda: ...