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

SO(3)-Irreducible Geometry in Complex Dimension Five and Ternary Generalization of Pauli Exclusion Principle