  We develop a complete local theory for CR embedded submanifolds of CR
manifolds in a way which parallels the Ricci calculus for Riemannian
submanifold theory. In particular, we establish the subtle relationship between
the submanifold and ambient standard tractor bundles, allowing us to relate the
respective normal Cartan (or tractor) connections via a CR Gauss formula. The
CR invariant tractor calculus of CR embeddings is developed concretely in terms
of the Tanaka-Webster calculus of an arbitrary (suitably adapted) ambient
contact form. This enables straightforward and explicit calculation of the
pseudohermitian invariants of the embedding which are also CR invariant. These
are extremely difficult to find and compute by more na\"ive methods. We
conclude by establishing a CR analogue of the classical Bonnet theorem in
Riemannian submanifold theory.
