Maslov Type Index for Lagrangian Paths

Abstract

Let $({\mathbb R}^{2n},\tilde \omega _0)$ be the standard symplectic space with $\tilde \omega _0=\displaystyle \sum _{i=1}^ndx_i\wedge dy_i$. For $z_i=(x_i,y_i)\in {\mathbb R}^n\times {\mathbb R}^n,\;i=1,2$, there holds

$$\displaystyle \tilde \omega _0(z_1,z_2)=\langle x_1,y_2\rangle -\langle x_2,y_1\rangle . $$

A Lagrangian subspace $L\subset ({\mathbb R}^{2n},\tilde \omega _0)$ is a dimensional n subspace with $\tilde \omega _0(z_1,z_2)=0$ for all z ₁, z ₂ ∈ Lt