The Moduli Space of Octonionic Bundles as a Subvariety of Orthogonal Bundles

Let X be a compact Riemann surface of genus g≥2. An octonionic bundle over X is a fiber bundle whose fiber is the non-associative algebra of complex octonions, equivalently a principal G2(C)-bundle, where G2(C) is the exceptional Lie group of automorphisms of the octonions. We prove that the natural inclusion G2(C)↪SO(7,C) induces a closed embedding of the moduli space MOct(X) into the moduli space MSO(7,C)(X) of SO(7,C)-bundles. We further analyze the normal bundle to this embedding, computing its rank as 7(g−1) and providing an explicit cohomological description of its fibers, which enables explicit computations of tangent spaces and provides a foundation for deformation theory. As applications of the embedding, we prove that the image is a closed irreducible subvariety not contained in the singular locus of the ambient space, and we derive the Whitney formula c(Tamb)=c(T)·c(N) relating the Chern classes of the tangent bundle of MOct(X), the pullback of the ambient tangent bundle, and the normal bundle over the smooth locus.