Definition II.1

Let ( P , ω ) 𝑃 𝜔 (P,\omega) ( italic_P , italic_ω ) be a symplectic manifold of dim P = 2 ⁹ n 𝑃 2 𝑛 P=2n italic_P = 2 italic_n , a manifold L 𝐿 L italic_L smoothly embedded by a map e : L → P normal-: 𝑒 normal-→ 𝐿 𝑃 e:L\to P italic_e : italic_L → italic_P is called a Lagrange submanifold of P 𝑃 P italic_P if the pull-back to L 𝐿 L italic_L of the symplectic form ω 𝜔 \omega italic_ω on P 𝑃 P italic_P by e 𝑒 e italic_e vanishes on L 𝐿 L italic_L

e * ⁹ ω = 0 , superscript 𝑒 𝜔 0 e^{*}\omega=0, italic_e start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_ω = 0 ,

and L 𝐿 L italic_L is of maximal possible dimension compatible with the symplectic structure ω 𝜔 \omega italic_ω , i.e. dim L = n 𝐿 𝑛 L=n italic_L = italic_n .

Definition II.5

Let ( P ^ , Îș ^ ) normal-^ 𝑃 normal-^ 𝜅 (\hat{P},\hat{\kappa}) ( ^ start_ARG italic_P end_ARG , ^ start_ARG italic_Îș end_ARG ) be a contact manifold of dimension 2 ⁹ n - 1 2 𝑛 1 2n-1 2 italic_n - 1 , a ( n - 1 ) 𝑛 1 (n-1) ( italic_n - 1 ) -dimensional manifold N 𝑁 N italic_N such that

e ^ * ⁹ Îș ^ = 0 , superscript ^ 𝑒 ^ 𝜅 0 \hat{e}^{*}\hat{\kappa}=0, ^ start_ARG italic_e end_ARG start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ^ start_ARG italic_Îș end_ARG = 0 ,

with e ^ : N → P ^ normal-: normal-^ 𝑒 normal-→ 𝑁 normal-^ 𝑃 \hat{e}:N\to\hat{P} ^ start_ARG italic_e end_ARG : italic_N → ^ start_ARG italic_P end_ARG an embedding, is called a Legendre submanifold of P ^ normal-^ 𝑃 \hat{P} ^ start_ARG italic_P end_ARG .