# Non-degenerate Poisson bracket and even-dimensional manifold

 P: 1,235 From this reference: http://www.amazon.com/exec/obidos/tg...glance&s=books titled From Classical to Quantum Mechanics, I quote the following: ( $$\xi^i$$ are coordinate functions) Let M be a manifold of dimension n. If we consider a non-degenerate Poisson bracket, i.e. such that $$\{\xi^i,\xi^j\} \equiv \omega^i^j$$ is an inversible matrix, we may define the inverse $$\omega_i_j$$ by requiring $$\omega_i_j \omega^j^k = \delta_i^k$$ We define a tensorial quantity $$\omega \equiv \frac{1}{2}\;\omega_i_j \; d\xi^i \wedge d\xi^j$$ which turns out to be a non-degenerate 2-form. This implies that the dimension of the manifold M is necessarily even. My questions are the following: I dont understand the two statement that I have put in red above. What is a non-degenerate 2-form? Why does this one above 'turns out' to be non-degenerate? Why does that imply that M is even? Additional comments would be welcome. Like concerning the meaning of $$\omega$$ above. In addition, I guess the point here by a shorter way: I think that all odd-dimensional antisymmetric matrices are singular. Is there a link with the language used above? Warm thanks in advance, Michel