Removing parametrized rays symplectically

DOI10.4310/JSG.2022.V20.N2.A4arXiv2009.05465MaRDI QIDQ6348912

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Publication date: 11 September 2020

Abstract: Extracting isolated rays from a symplectic manifold result in a manifold symplectomorphic to the initial one. The same holds for higher dimensional parametrized rays under an additional condition. More precisely, let

(M, o m e g a)

be a symplectic manifold. Let

[0, i n f t y) i m e s Q s u b s e t m a t h b b R i m e s Q

be considered as parametrized rays

[0, i n f t y)

and let

v a r p h i : [- 1, i n f t y) i m e s Q o M

be an injective, proper, continuous map immersive on

(- 1, i n f t y) i m e s Q

. If for the standard vector field

f r a c p a r t i a l p a r t i a l t

on

m a t h b b R

and any further vector field

u

tangent to

(- 1, i n f t y) i m e s Q

the equation

v a r p h i^{*} o m e g a (f r a c p a r t i a l p a r t i a l t, u) = 0

holds then

M

and

M s e t m i n u s v a r p h i ([0, i n f t y) i m e s Q)

are symplectomorphic.

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