Let \(\alpha\) and \(\beta\) be complex vector bundles over a closed connected smooth \(n\)-manifold \(M\). The author employs the singularity method (in the spirit of his book [Vector fields and other vector bundle morphisms – a singularity approach, Lect. Notes Math. 847 (1981; Zbl 0459.57016)]) to compare complex and real monomorphisms from \(\alpha\) to \(\beta\). Two such monomorphisms are considered as equal if they are regularly homotopic.
The existence and classification results in the complex and in the real setting are related by transition homomorphisms of normal bordism groups; these homomorphisms fit into a long exact sequence of Gysin type. Under suitable conditions, explicit calculations (giving results in terms of characteristic classes) are possible. For instance, if \(\dim _\mathbb R(\alpha)=2\), \(\dim _\mathbb R(\beta)=n\), and \(M\) is nonorientable, then the following statements are proved to be equivalent: (i) there exists a complex monomorphism \(u:\alpha\rightarrow \beta\); (ii) there exists a real monomorphism \(u:\alpha\rightarrow \beta\); and (iii) the Stiefel-Whitney class \(w_n(\beta -\alpha)= \sum_{i\geq 0} w_2(\alpha)^{i} w_{n-2i}(\beta)\) vanishes.
The paper contains many examples (mostly with \(M\) being a product of spheres and/or complex projective spaces) illustrating interesting phenomena. For instance, it may happen that infinitely many real monomorphisms exist where there is no complex monomorphism \(\alpha\rightarrow\beta\).