Gauss-Bonnet formula is used to derive a new and simple theorem of nonexist
ence of vacuum static nonsingular Lorentzian wormholes. We also derive simp
le proofs for the nonexistence of Lorentzian wormhole solutions for some cl
asses of static matter such as, for instance, real scalar fields with a gen
eric potential obeying phi V'(phi) greater than or equal to 0 and massless
fermion fields.