regarding metric and symplectic forms - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T16:21:46Z http://mathoverflow.net/feeds/question/62719 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/62719/regarding-metric-and-symplectic-forms regarding metric and symplectic forms tomate 2011-04-23T09:11:06Z 2011-05-07T20:22:14Z <p>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</p> <p>$h = g + c[g,\mathbb{J}],$</p> <p>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,</p> <p>$h>0$</p> <p>?</p> http://mathoverflow.net/questions/62719/regarding-metric-and-symplectic-forms/62768#62768 Answer by Giuseppe for regarding metric and symplectic forms Giuseppe 2011-04-23T19:21:57Z 2011-04-23T20:11:22Z <p>I have met the combination of complex structures, symplectic forms, scalar products on the same real vector space in the following context. </p> <p>Let $V$ be a real vector space.<br> The complex vector space structures compatible with the assigned real vector space structure of $V$ are in correspondence one-to-one with the <em>complex operators</em> 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 $v\in V$.<br> Obviously such structures $(V,\mathbb{J})$ exist if and only if $\dim{V}$ is pair.</p> <p><strong>Definition</strong>. 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 $\Im\eta=\Omega$.</p> <p><strong>Theorem</strong> $\mathbb{J}$ 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 $\eta=(\Omega^\flat\circ\mathbb{J})^\sharp+i\Omega$.</p>