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It is supposed the existence of a curved graphene sheet with the geometry ofa Bour surface $B_{n}$, such as the catenoid (or helicoid), $B_{0}$, and theclassical Enneper surface, $B_{2}$, among others. In particular, in this work,the propagation of the electronic degrees of freedom on these surfaces isstudied based on the Dirac equation. As a consequence of the polar geometry of$B_{n}$, it is found that the geometry of the surface causes the Dirac fermionsto move as if they would be subjected to an external potential coupled to aspin-orbit term. The geometry-induced potential is interpreted as a barrierpotential, which is asymptotically zero. Furthermore, the behaviour ofasymptotic Dirac states and scattering states are studied through theLippmann-Schwinger formalism. It is found that for surfaces $B_{0}$ and$B_{1}$, the total transmission phenomenon is found for sufficiently largevalues of energy, while for surfaces $B_{n}$, with $n\geq 2$, it is shown thatthere is an energy point $E_{K}$ where Klein's paradox is realized, while forenergy values $E\gg E_{K}$ it is found that the conductance of the hypotheticalmaterial is completely suppressed, $\mathcal{G}(E)\to 0$.

Klein paradox between transmitted and reflected Dirac waves on Bour surfaces