A CR manifold is a real smooth manifold 
with a complex subbundle 
of its complexified tangent
bundle satisfying 
and closed under Lie brackets of local sections.
Its complex rank is the CR dimension. The CR codimension
is 
.
A smooth real hypersurface in 
inherits a CR structure of dimension 
from the complex tangent directions. A CR function is a
function annihilated by all local bundle
sections of 
.
A local CR embedding into complex Euclidean space is a smooth embedding
for which the given CR structure agrees with the CR structure induced on its image
(Fritsch and Heinzner 2022, p. 140).
An abstract CR structure need not admit such a local embedding. The Trautman conjecture proposed a sufficient condition in real dimension 3 in terms of a nowhere-zero closed canonical form (Curry 2026).
