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Suppose $(M,\omega)$ is a symplectic manifold and $L \subset M$ is a compact Lagrangian submanifold. Is it the case that the first Chern class $c_1(TM) \in H^2(M)$ vanishes when restricted to the Lagrangian submanifold $L$?

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Yes, if you ignore 2-torsion: After all, if you give $TM$ a complex structure compatible with $\omega$, say $J$, then you'll have that $TM$ pulls back to $L$ to become isomorphic to $TL\oplus T^*L\simeq TL\oplus J(TL)$, so, as a complex bundle, $TM$ pulls back to be the complexification of $TL$, and hence $$L^*\bigl(c_1(TM)\bigr) = c_1(TL^\mathbb{C}) = -c_1(TL^\mathbb{C}),$$ since $TL^\mathbb{C}$ is isomorphic to its complex conjugate bundle. Thus $$ 2\,L^*\bigl(c_1(TM)\bigr) =0. $$

(Note that I use $L^*$ to denote the pullback (aka restriction) map $H^*(M)\to H^*(L)$ in cohomology.)

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