The problem of understanding CR geometries embedded as submanifolds in

higher dimensional CR manifolds arises in higher dimensional complex

analysis, including the study of singularities of analytic

varieties. It has also been studied intensively in connection with

rigidity questions. Despite considerable earlier work the local theory

has not been fully understood.

We develop from scratch a CR invariant local theory based on CR

tractor calculus (i.e. the associated bundle). This produces the tools

for constructing local invariants and invariant operators in a way

parallel to the classical Gauss-Codazzi-Ricci calculus for Riemannian

submanifolds. It also enables a practical and conceptual approach to

a Bonnet Theorem and potentially the rigidity questions.

This is joint work with Rod Gover.