The conventional Lagrangian density with the minimal coupling between the spinor fields and the non-Abelian gauge fields in a Minkowskian space is transformed to that in the Riemannian space by the concept of geodesic corrdinates with a point at infinity as a pole.
In this works the effect of the gauge invariance of the Lagrangian density on the geometry is examined. That is, if we identify $R_{αβγδ}$ with k $J_{αβγδ}$ from the similarity of their tensorial characters and assume the space and the spinor fields to satisfy
$\frac{δj^α_k}{δs}=0$ for all k
than the space of the spinor fields results in a recurrent space.
Also the cases of the strong interactions the weak interactions and the electromagnetic interactions are discussed. Especially for the electromagnetic interaction the space is reduced to a Minkowskian space, which agrees with the fact that the quantum electrodynamics works well in a Minkowskian space.
