  In this paper, we aim to provide a notion of "relative objects", i.e. objects
equipped with some sort of subobjects, in differential topology. In spite of
active researches relating them, e.g. knot theory or the theory of manifolds
with corners, there seem to be poor general notions to deal with them.
Moreover, we want even more direct differential calculus on relative objects
and extension of classical notions and theories to relative situations; e.g.
functions, vector fields, jet bundles, singularities, and so on. To establish
this, the notion of arrangements of manifolds is introduced, which, for
example, enables us to control behaviors of smooth maps on manifolds around
corners. We construct jet bundles and prove a relative version of
Transversality Theorem for some sorts of arrangements. Finally, embedding
theorem of manifolds with faces into polyhedra is proved as an application.
