Maslov Type Index for Lagrangian Paths

Chungen Liu
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Chungen Liu: Guangzhou University, School of Mathematical and Information Science

Chapter Chapter 6 in Index theory in nonlinear analysis, 2019, pp 161-176 from Springer

Abstract: Abstract Let ( ℝ 2 n , ω ~ 0 ) $$({\mathbb R}^{2n},\tilde \omega _0)$$ be the standard symplectic space with ω ~ 0 = ∑ i = 1 n d x i ∧ d y i $$\tilde \omega _0=\displaystyle \sum _{i=1}^ndx_i\wedge dy_i$$ . For z i = ( x i , y i ) ∈ ℝ n × ℝ n , i = 1 , 2 $$z_i=(x_i,y_i)\in {\mathbb R}^n\times {\mathbb R}^n,\;i=1,2$$ , there holds ω ~ 0 ( z 1 , z 2 ) = 〈 x 1 , y 2 〉 − 〈 x 2 , y 1 〉 . $$\displaystyle \tilde \omega _0(z_1,z_2)=\langle x_1,y_2\rangle -\langle x_2,y_1\rangle . $$ A Lagrangian subspace L ⊂ ( ℝ 2 n , ω ~ 0 ) $$L\subset ({\mathbb R}^{2n},\tilde \omega _0)$$ is a dimensional n subspace with ω ~ 0 ( z 1 , z 2 ) = 0 $$\tilde \omega _0(z_1,z_2)=0$$ for all z 1, z 2 ∈ Lt

Date: 2019
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DOI: 10.1007/978-981-13-7287-2_6

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