  We propose a notion of a ternary skew-symmetric covariant tensor of 3rd
order, consider it as a 3-dimensional matrix and study a ten-dimensional
complex space of these tensors. We split this space into a direct sum of two
five-dimensional subspaces and in each subspace there is an irreducible
representation of the rotation group SO(3) -> SO(5). We find two independent
SO(3)-invariants of ternary skew-symmetric tensors, where one of them is the
Hermitian metric and the other is the quadratic form. We find the stabilizer of
this quadratic form and its invariant properties. Making use of these invariant
properties we define a SO(3)-irreducible geometric structure on a
five-dimensional complex Hermitian manifold. We study a connection on a
five-dimensional complex Hermitian manifold with a SO(3)-irreducible geometric
structure, find its curvature and torsion. The structures proposed in this
paper and their study are motivated by a ternary generalization of the Pauli's
principle proposed by R. Kerner.
