  The entire geometric formulations of the BRST and the anti-BRST structures
are worked out in presence of the Nakanishi-Lautrup field. It is shown that in
the general form of gauge fixing mechanisms within the Faddeev-Popov
quantization approach, the antiBRST invariance reflects thoroughly the
classical symmetry of the Yang-Mills theories with respect to gauge fixing
methods. The Nakanishi-Lautrup field is also defined and worked out as a
geometric object. This formulation helps us to introduce two absolutely new
topological invariants of quantized Yang-Mills theories, so called the
Nakanishi-Lautrup invariants. The cohomological structure of the anti-BRST
symmetry is also studied and the anti-BRST topological index is derived
accordingly.
