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Why Lagrangians of two (real) torus are lines?

1

$begingroup$

I am reading an article on Mirror Symmetry, where an example is given : the two (real) dimensional torus.

My question is a basic one : taking the symplectic form (if ones focuses on the symplectic aspect) to be the area form, the author says that Lagrangians (submanifolds closed by this form) are just lines.

Is this simply because lines are the zero area submanifolds?
Why there aren't more than these?

Thank you.

manifolds symplectic-geometry mirror-symmetry

share|cite|improve this question

edited Mar 25 at 10:08

Andrews

1,2962423

asked Sep 20 '18 at 20:36

vanmerivanmeri

709

$endgroup$

add a comment | 

1

$begingroup$

I am reading an article on Mirror Symmetry, where an example is given : the two (real) dimensional torus.

My question is a basic one : taking the symplectic form (if ones focuses on the symplectic aspect) to be the area form, the author says that Lagrangians (submanifolds closed by this form) are just lines.

Is this simply because lines are the zero area submanifolds?
Why there aren't more than these?

Thank you.

manifolds symplectic-geometry mirror-symmetry

share|cite|improve this question

edited Mar 25 at 10:08

Andrews

1,2962423

asked Sep 20 '18 at 20:36

vanmerivanmeri

709

$endgroup$

add a comment | 

$begingroup$

I am reading an article on Mirror Symmetry, where an example is given : the two (real) dimensional torus.

My question is a basic one : taking the symplectic form (if ones focuses on the symplectic aspect) to be the area form, the author says that Lagrangians (submanifolds closed by this form) are just lines.

Is this simply because lines are the zero area submanifolds?
Why there aren't more than these?

Thank you.

manifolds symplectic-geometry mirror-symmetry

share|cite|improve this question

edited Mar 25 at 10:08

Andrews

1,2962423

asked Sep 20 '18 at 20:36

vanmerivanmeri

709

$endgroup$

share|cite|improve this question

edited Mar 25 at 10:08

Andrews

1,2962423

asked Sep 20 '18 at 20:36

vanmerivanmeri

709

share|cite|improve this question

edited Mar 25 at 10:08

Andrews

1,2962423

asked Sep 20 '18 at 20:36

vanmerivanmeri

709

edited Mar 25 at 10:08

Andrews

1,2962423

edited Mar 25 at 10:08

Andrews

1,2962423

asked Sep 20 '18 at 20:36

vanmerivanmeri

709

asked Sep 20 '18 at 20:36

vanmerivanmeri

709