Harmonicity conditions for nonsymmetric metric

Algebraic differential properties of the nonsymmetric metric are considered. It is shown that the nonsymmetric metric is characterized by a pair of vector fields and a pair of covector fields connected by two identities. If these fields are zero (just in this way the harmonicity conditions are formulated), the nonsymmetric metric is called harmonic. An example of the harmonic metric is any metric obeying the Born-Infeld equations. The case of the symmetric metric is trivial