Lagrangian displacement (symplectic geometry)

A Lagrangian $L\subset M$ is displaceable by a class of isotopies when some isotopy ${\varphi }_t$ in that class satisfies ${\varphi }_1(L)\cap L=\varnothing$.

For a compact orientable Lagrangian $L$, the identification $NL\cong T^{\ast }L$ givesIf $\chi (L)\ne 0$, no smoothly isotopic copy of $L$ can be disjoint from it.

A simple closed curve dividing a symplectic two-sphere into two regions of equal area cannot be displaced by a symplectic isotopy. Any disjoint image would lie in one complementary disc, yet it would still have to bound a disc of half the total area.