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totally real submanifold
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(Definition)
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Definition 1 Suppose that $M \subset {\mathbb{C}}^N$ is a CR submanifold. If the CR dimension of $M$ is 0, we say that $M$ is totally real. If in addition $M$ is generic, then $M$ is said to be maximally totally real (or sometimes just maximally real).
Note that if $M$ is maximally totally real, then the real dimension is automatically $N$ this is because $T_x^c(M) = T_x(M) \cap JT_x(M)$ (the complex tangent space) is of dimension 0, and thus $T_x(M)$ must be of real dimension $N$ if $M$ is to be a generic manifold.
- 1
- M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. Real Submanifolds in Complex Space and Their Mappings, Princeton University Press, Princeton, New Jersey, 1999.
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"totally real submanifold" is owned by jirka.
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See Also: CR submanifold, generic manifold
| Other names: |
totally real manifold |
| Also defines: |
maximally totally real submanifold, maximally totally real manifold, maximally real manifold |
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Cross-references: generic manifold, complex tangent space, dimension, addition, real, CR dimension, CR submanifold
This is version 2 of totally real submanifold, born on 2005-01-06, modified 2005-03-07.
Object id is 6624, canonical name is TotallyRealSubmanifold.
Accessed 5211 times total.
Classification:
| AMS MSC: | 32V05 (Several complex variables and analytic spaces :: CR manifolds :: CR structures, CR operators, and generalizations) |
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Pending Errata and Addenda
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