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CR Embedded Submanifolds of CR Manifolds

About this Title

Sean N. Curry and A. Rod Gover

Publication: Memoirs of the American Mathematical Society
Publication Year: 2019; Volume 258, Number 1241
ISBNs: 978-1-4704-3544-8 (print); 978-1-4704-5073-1 (online)
Published electronically: March 11, 2019
Keywords:CR embeddings, CR invariants, tractor calculus, Gauss-Codazzi-Ricci equations, Bonnet theorem.

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Weighted Tanaka-Webster Calculus
  • Chapter 3. CR Tractor Calculus
  • Chapter 4. CR Embedded Submanifolds and Contact Forms
  • Chapter 5. CR Embedded Submanifolds and Tractors
  • Chapter 6. Higher Codimension Embeddings
  • Chapter 7. Invariants of CR Embedded Submanifolds
  • Chapter 8. A CR Bonnet Theorem


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. We define a normal tractor bundle in the ambient standard tractor bundle along the submanifold and show that the orthogonal complement of this bundle is not canonically isomorphic to the standard tractor bundle of the submanifold. By determining the subtle relationship between submanifold and ambient CR density bundles we are able to invariantly relate these two tractor bundles, and hence to invariantly relate the normal Cartan connections of the submanifold and ambient manifold by a tractor analogue of the Gauss formula. This leads also to CR analogues of the Gauss, Codazzi, and Ricci equations. The tractor Gauss formula includes two basic invariants of a CR embedding which, along with the submanifold and ambient curvatures, capture the jet data of the structure of a CR embedding. These objects therefore form the basic building blocks for the construction of local invariants of the embedding. From this basis we develop a broad calculus for the construction of the invariants and invariant differential operators of CR embedded submanifolds.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 naïve methods. We conclude by establishing a CR analogue of the classical Bonnet theorem in Riemannian submanifold theory.

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