Geometry of fibered graphs of mappings

In this paper, we examine the differential-geometric aspect of constant-rank mappings of smooth manifolds based on the concept of a graph as a smooth submanifold in the space of the direct product of the original manifolds. The nonmaximality of the rank provides the fibered nature of the graph. A Riemannian structure on manifolds enriches the geometry of the graph, which now essentially depends on the induced field of the metric tensor; we characterize relatively affine, projective, and $g$-umbilical mappings. The final part of the paper is devoted to mappings of Euclidean spaces of the types described earlier in terms of V. T. Bazylev's constructive graph.