Differential Geometry of Composite Fibred Manifolds

In classical field theory, the composite fibred manifolds Y -> Z -> X provides the adequate mathematical formulation of gauge models with broken symmetries, e.g., the gauge gravitation theory. This work is devoted to connections on composite fibred manifolds. In particular, we get the horizontal splitting of the vertical tangent bundle of a composite fibred manifold, besides the familiar one of its tangent bundle. This splitting defines the modified covariant differential and implies the special fashion of Lagrangian densities of field models on composite manifolds. The spinor composite bundles are examined.