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CR Embedded Submanifolds of CR Manifolds

Sean N. Curry The University of Auckland, Auckland, New Zealand
A. Rod Gover The University of Auckland, Auckland, New Zealand
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Softcover ISBN: 978-1-4704-3544-8
Product Code: MEMO/258/1241
List Price: $81.00 MAA Member Price:$72.90
AMS Member Price: $48.60 Electronic ISBN: 978-1-4704-5073-1 Product Code: MEMO/258/1241.E List Price:$81.00
MAA Member Price: $72.90 AMS Member Price:$48.60
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List Price: $121.50 MAA Member Price:$109.35
AMS Member Price: $72.90 Click above image for expanded view CR Embedded Submanifolds of CR Manifolds Sean N. Curry The University of Auckland, Auckland, New Zealand A. Rod Gover The University of Auckland, Auckland, New Zealand Available Formats:  Softcover ISBN: 978-1-4704-3544-8 Product Code: MEMO/258/1241  List Price:$81.00 MAA Member Price: $72.90 AMS Member Price:$48.60
 Electronic ISBN: 978-1-4704-5073-1 Product Code: MEMO/258/1241.E
 List Price: $81.00 MAA Member Price:$72.90 AMS Member Price: $48.60 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$121.50 MAA Member Price: $109.35 AMS Member Price:$72.90
• Book Details

Memoirs of the American Mathematical Society
Volume: 2582019; 81 pp
MSC: Primary 32; 53;

The authors develop a complete local theory for CR embedded submanifolds of CR manifolds in a way which parallels the Ricci calculus for Riemannian submanifold theory. They 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 the authors 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 also leads 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 the authors 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. The authors conclude by establishing a CR analogue of the classical Bonnet theorem in Riemannian submanifold theory.

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

• Requests

Review Copy – for reviewers who would like to review an AMS book
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Volume: 2582019; 81 pp
MSC: Primary 32; 53;

The authors develop a complete local theory for CR embedded submanifolds of CR manifolds in a way which parallels the Ricci calculus for Riemannian submanifold theory. They 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 the authors 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 also leads 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 the authors 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. The authors conclude by establishing a CR analogue of the classical Bonnet theorem in Riemannian submanifold theory.

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