Lagrangian subbundle
= Lagrangian subbundle
{c}
A rank-$n$ <vector subbundle> of a rank-$2n$ <symplectic vector bundle> is Lagrangian when each fibre is a <Lagrangian subspace>. If it exists, a <compatible almost complex structure> supplies a complementary <Lagrangian subbundle> by applying $J$. Existence is a global condition: the area-oriented <tangent bundle> of the two-sphere has no real line subbundle, by triviality of real <line bundles> over the sphere and the <Hairy ball theorem>.