CR manifold

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a CR manifold, or Cauchy–Riemann manifold, is a <a href="/facts/Differentiable_manifold/aSnbXKcV">differentiable manifold</a> together with a geometric structure modeled on that of a real <a href="/facts/Hypersurface/cTvtLVsV">hypersurface</a> in a <a href="/facts/Real_number/R02gw5Pb">complex vector space</a>, or more generally modeled on an <a href="/facts/Edge-of-the-wedge_theorem/pVPWPC42">edge of a wedge</a>.
Formally, a CR manifold is a differentiable manifold M together with a preferred complex distribution L, or in other words a complex <a href="/facts/Subbundle/KFWrEleF">subbundle</a> of the <a href="/facts/Complexified/fDfFLccF">complexified</a> <a href="/facts/Tangent_bundle/dq8zW9Kb">tangent bundle</a> 
 
 
 
 
 C
 
 T
 M
 =
 T
 M
 
 ⊗
 
 
 R
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} TM=TM\otimes _{\mathbb {R} }\mathbb {C} }
 
 such that

<ul><li>
 
 
 
 [
 L
 ,
 L
 ]
 ⊆
 L
 
 
 {\displaystyle [L,L]\subseteq L}
 
 (L is formally <a href="/facts/Frobenius_integration_theorem/A8gu0fjt">integrable</a>)</li>
<li>
 
 
 
 L
 ∩
 
 
 
 L
 ¯
 
 
 
 =
 {
 0
 }
 
 
 {\displaystyle L\cap {\bar {L}}=\{0\}}
 
.</li></ul>
The subbundle L is called a CR structure on the manifold M.
The abbreviation CR stands for "<a href="/facts/Cauchy%E2%80%93Riemann/SWkIHflU">Cauchy–Riemann</a>" or "Complex-Real".

CR manifold open-in-new

CR manifold