# 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