| Summary: | Motivated by a ternary generalization of the Pauli exclusion principle proposed by R. Kerner, we propose a notion of a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>3</mn></msub></semantics></math></inline-formula>-skew-symmetric covariant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>SO</mi></mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula>-tensor of the third order, consider it as a 3-dimensional matrix, and study the geometry of the 10-dimensional complex space of these tensors. We split this 10-dimensional space into a direct sum of two 5-dimensional subspaces by means of a primitive third-order root of unity <i>q</i>, and in each subspace, there is an irreducible representation of the rotation group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>SO</mi></mrow><mo>(</mo><mn>3</mn><mo>)</mo><mo>↪</mo><mrow><mi>SU</mi></mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></semantics></math></inline-formula>. We find two <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>SO</mi></mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula>-invariants of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>3</mn></msub></semantics></math></inline-formula>-skew-symmetric tensors: one is the canonical Hermitian metric in five-dimensional complex vector space and the other is a quadratic form denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>(</mo><mi>z</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We study the invariant properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>(</mo><mi>z</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> and find its stabilizer. Making use of these invariant properties, we define an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>SO</mi></mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula>-irreducible geometric structure on a five-dimensional complex Hermitian manifold. We study a connection on a five-dimensional complex Hermitian manifold with an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>SO</mi></mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula>-irreducible geometric structure and find its curvature and torsion.
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