Parallel immersions into spaces of constant curvature and conformal transformations (Q2733978)

The notion of exterior parallelism of immersions into Euclidean space introduced by \textit{H. R. Farran} and \textit{S. A. Robertson} [J. Lond. Math. Soc. (2) 35, 527-538 (1987; Zbl 0623.53022)] is extended here to the case when the target is a manifold of constant sectional curvature and some results on different types of parallel ranks are generalized as well. By using \textit{K. Yano} and \textit{S. Ishihara} [J. Differ. Geom. 6, 95-118 (1971; Zbl 0222.53052)] and \textit{B. Wegner} [J. Differ. Geom. 16, 93-100 (1981; Zbl 0406.53005)] some characterizations are obtained in terms of the curvature of certain subbundles of the normal bundle. Particular implications for the locus of the focal points of parallel rank of an immersion are shown. Some conditions under which parallel rank of an immersion is invariant under the conformal transformations between spaces of constant curvature are obtained at the end.