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.

单词	TotallyRealSubmanifold
释义	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.
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