Recent theoretical and experimental studies suggest that SmB$_6$ is the first topological Kondo insulator: A material in which the interaction between localized and itinerant electrons renders the bulk insulating at low temperature, while topological surface states leave the surface metallic. While this would elegantly explain the materials puzzling conductivity, we find the experimentally observed candidates for both predicted topological surface states to be of trivial character instead: The surface state at $bar{Gamma}$ is very heavy and shallow with a mere $sim 2$ meV binding energy. It exhibits large Rashba splitting which excludes a topological nature. We further demonstrate that the other metallic surface state, located at $bar{X}$, is not an independent in-gap state as supposed previously, but part of a massive band with much higher binding energy (1.7 eV). We show that it remains metallic down to 1 K due to reduced hybridization with the energy-shifted surface 4$f$ level.
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