  The altermagnetism influences the electronic states allowing the presence of
non-relativistic spinsplittings. Since altermagnetic spin-splitting is present
along specific k-paths of the 3D Brillouin zone, we expect that the
altermagnetic surface states will be present on specific surface orientations.
We unveil the properties of the altermagnetic surface states considering three
representative space groups: tetragonal, orthorhombic and hexagonal. We
calculate the 2D projected Brillouin zone from the 3D Brillouin zone. We study
the surfaces with their respective 2D Brillouin zones establishing where the
spin-splittings with opposite sign merge annihilating the altermagnetic
properties and on which surfaces the altermagnetism is preserved. Looking at
the three principal surface orientations, we find that for several cases two
surfaces are blind to the altermagnetism, while the altermagnetism survives for
one surface orientation. Which surface preserves the altermagnetism depends
also on the magnetic order. We show that an electric field orthogonal to the
blind surface can activate the altermagnetism. Our results predict which
surfaces to cleave in order to preserve altermagnetism in surfaces or
interfaces and this paves the way to observe non-relativistic altermagnetic
spin-splitting in thin films via spin-resolved ARPES and to interface the
altermagnetism with other collective modes. We open future perspectives for the
study of altermagnetic effects on the trivial and topological surface states.
