totally real submanifold

Definition.

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 (http://planetmath.org/GenericManifold), 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.

References

1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.

Title	totally real submanifold
Canonical name	TotallyRealSubmanifold
Date of creation	2013-03-22 14:56:05
Last modified on	2013-03-22 14:56:05
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	5
Author	jirka (4157)
Entry type	Definition
Classification	msc 32V05
Synonym	totally real manifold
Related topic	CRSubmanifold
Related topic	GenericManifold
Defines	maximally totally real submanifold
Defines	maximally totally real manifold
Defines	maximally real manifold