regarding metric and symplectic forms

2011-04-23T09:11:06Z

Let $g$ be a positive definite symmetric form on a $2n$-dimensional vector space (a metric), $\mathbb{J}$ the symplectic unit and $c$ a real number. Define the symmetric form

$h = g + c[g,\mathbb{J}],$

where $[\cdot,\cdot]$ is the commutator. Have you ever encountered this or similar objects? If so, what is the geometrical meaning of requiring it to be positive definite,

$h>0$

Answer by Giuseppe for regarding metric and symplectic forms

2011-04-23T19:21:57Z

I have met the combination of complex structures, symplectic forms, scalar products on the same real vector space in the following context.

Let $V$ be a real vector space.
The complex vector space structures compatible with the assigned real vector space structure of $V$ are in correspondence one-to-one with the complex operators on $V$, i.e. the linear operators $J$ on $V$ such that $\mathbb{J}^2=-id_V\equiv \mathbb{I}$; this correspondence is realized through the relation $(a+ib).v=av+b\mathbb{J}v$, for any $a,b\in\mathbb{R}$, and $v\in V$.
Obviously such structures $(V,\mathbb{J})$ exist if and only if $\dim{V}$ is pair.

Definition. Let $\Omega$ be a symplectic form and $\mathbb{J}$ a complex operator on $V$. The complex operator $\mathbb{J}$ is said to be adapted to $\Omega$ when there exists a pseudo-hermitian form $\eta$ on the complex vector space $(V,\mathbb{J})$ such that $\Im\eta=\Omega$.

Theorem $\mathbb{J}$ is adapted to $\Omega$ if and only if $\mathbb{J}$ is an isomorphism of $(V,\Omega)$, i.e. $\mathbb{J}^T\circ\Omega^\flat\circ\mathbb{J}=\mathbb{J}\circ\Omega\in L(V,V^*)$; and in the affirmative case there is a unique hermitian form $\eta$ on $(V,\Omega)$ with $\Im\eta=\Omega$, it is given by $\eta=(\Omega^\flat\circ\mathbb{J})^\sharp+i\Omega$.

regarding metric and symplectic forms - MathOverflow

regarding metric and symplectic forms

Answer by Giuseppe for regarding metric and symplectic forms