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The b-boundary is a mathematical tool used to attach a topological boundary to incomplete Lorentzian manifolds using a Riemaniann metric called the Schmidt metric on the frame bundle. In this paper we give the general form of the Schmidt metric in the case of Lorentzian surfaces. Furthermore, we write the Ricci scalar of the Schmidt metric in terms of the Ricci scalar of the Lorentzian manifold and give some examples. Finally, we discuss some applications to general relativity.