# totally real submanifold

###### Definition.

Suppose that $M\subset {\u2102}^{N}$ 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 ${T}_{x}^{c}(M)={T}_{x}(M)\cap J{T}_{x}(M)$ (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 |