We define a real linear mapping such that for any we have
Where is referred to as the complex structure on Note that that is applying twice we just negate the vector.
Let be the tangent space of at the point (that is, those vectors of which are tangent to ).
Do note that the complex tangent space is a real (not complex) vector space, despite its rather unfortunate name.
Let and be the complexified vector spaces, by just allowing the coefficents of the vectors to be complex numbers. That is for we allow and to be complex numbers. Next we can extend the mapping to be -linear on these new vector spaces and still get that as before. We notice that the operator has two eigenvalues, and .
The submanifold is called a CR submanifold (or just CR manifold) if the dimension of is constant for all The complex dimension of will then be called the CR dimension of
Note that sometimes is written as and referred to as the space of antiholomorphic vectors, where an antiholomorphic vector is a tangent vector which can be written in terms of the basis
The CR in the name refers to Cauchy-Riemann and that is because the vector space corresponds to differentiating with respect to
- 1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.
- 2 Albert Boggess. , CRC, 1991.
|Date of creation||2013-03-22 14:49:04|
|Last modified on||2013-03-22 14:49:04|
|Last modified by||jirka (4157)|
|Defines||CR vector field|
|Defines||complex tangent space|
|Defines||space of antiholomorphic vectors|