The author applies a notion of conjugate connection introduced by S. Kobayashi and E. Shinozaki to study the geometry of 4-manifolds dealing exclusively with the simplest compact simple Lie group \(SU(3)\). In this case the outer automorphism group is isomorphic to the cyclic group \(\mathbb{Z}_2\). One of the aims of this paper is to prove a fixed point theorem under the action of \(\mathbb{Z}_2\) on the irreducible part \({\mathcal B}^{*}(P)\) of the quotient space \({\mathcal B}(P)\) of gauge-equivalent connections on a reducible principal \(SU(3)\)-bundle \(P\) along the irreducible part \({\mathcal B}^{*}(Q)\) of the quotient spaces \({\mathcal B}(Q)\) of \(SO(3)\)-subbundles \(Q\) of \(P\). Following a line suggested by Donaldson, the author defines simple \(SU(3)\)-instanton invariants \(q_k(X)\) for certain smooth 4-manifolds.