Articles of symplectic geometry

Do Hamiltonian Vector Fields Correspond to Co-Closed 1-Forms on a Closed, Connected Riemannian Manifold (in a Meangingful Sense of the Phrase)?

Let $(M^n, g)$ be a closed, connected Riemannian manifold, and let $T^*M$ be its cotangent bundle. Let $\tilde{g}$ be a Riemannian metric on $T^*M$. Let $\omega$ be the canonical symplectic 2-form on $T^*M$. There are 3 isomorphisms associated to $(T^*M, \tilde{g})$: (1) the “musical isomorphism” given by the Riemannian metric $\tilde{g}$ from $\Gamma(T^*M)$ to $\Omega^1(T^*M)$, […]

Inclusion $O(2n)/U(n)\to GL(2n,\mathbb{R})/GL(n,\mathbb{C}) $

How to show, that an inclusion of homogenious spaces $$O(2n)/U(n)\to GL(2n,\mathbb{R})/GL(n,\mathbb{C}) $$ is homotopy equivalence? The big space is the space of complex structures on $\mathbb{R}^{2n}$. This is an exercise in the book of McDuff and Salamon on symplectic topology, and they recommend to use polar decomposition, but it gives a deformational retraction $GL(2n,\mathbb{R}) \to […]

Canonical symplectic form on cotangent bundle of complex manifold

Given any smooth manifold $M$, its cotangent bundle $T^*M$ is a symplectic manifold, with the canonical symplectic form. If $M$ is a complex manifold then $T^*M$ is also a complex manifold. Thus, $T^*M$ is a complex symplectic manifold. Does it follow that the canonical symplectic form is holomorphic? If not, what condition can be placed […]

Kähler form convention

I’ve been wondering about this for a while and I have my ideas about the answer, but I would like to make sure once and for all that I’m not missing something. Let’s look at this from a purely linear algebra perspective: let $h$ be a Hermitian inner product on a complex vector space. Should […]

Symplectic basis $(A_i,B_i)$ such that $S= $ span$(A_1,B_1,…,A_k,B_k)$ for some $k$ when $S$ is symplectic

Let $(V,\omega)$ be a symplectic vector space of dimension $2n$. How can I show that for a symplectic subspace $S \subset V$, there exists a symplectic basis $(A_i,B_i)$ such that $S= $ span$(A_1,B_1,…,A_k,B_k)$ for some $k$. I know that since $S$ is symplectic, $S \cap S^\perp= \lbrace0 \rbrace$ and this is true iff $\omega|_S$ is […]

Is the symplectic group $\operatorname{Sp}(2n,\mathbb{R})$ simple?

Is the symplectic group $\operatorname{Sp}(2n,\mathbb{R})$ simple? Wikipedia states that the Lie algebra $\mathfrak{sp}(2n,\mathbb{R})$ is simple. However it only lists projective groups in its list of simple Lie groups.

How to show that geodesics exist for all of time in a compact manifold?

Let $M$ be a compact manifold and the tangent bundle $TM$ have a Riemannian metric $g$ so that it is isomorphic to the cotangent bundle $T^*M$. Consider the pull-back of the canonical symplectic form $\omega_{std}$ to define a symplectic form $\omega_g$ on $TM. $ Assuming the following theorem: The geodesics in $M$ are the images […]

Trouble understanding proof of this proposition on contact type hypersurfaces

I have finally started working on my thesis. I am reading the first reference book, which is Dusa McDuff, D. Salamon, Introduction to Symplectic Topology. It’s a tough struggle, given my not-too-great experience with differential forms. I will recall a few concepts. A manifold $M$ is called symplectic if it is equipped with a closed […]

Exact sequence arising from symplectic manifold

Let $M$ be a symplectic manifold, why is the following sequence exact? $$0\to \mathbb{R} \to C^\infty (M)\to A\to 0$$ Here $A$ is the set of global Hamiltonian vector fields.

volume of projective space $\text{Vol}(\mathbb CP^N)$

How can we compute the volume of projective space $$\text{Vol}(\mathbb CP^N)$$