A CR manifold is a C∞ differentiable manifold endowed with a complex subbundle T1,0(M) of the complexified tangent bundle T (M)⊗C satisfying T1,0(M)∩T1,0(M) = (0) and the Frobenius (formal) integrability property
∞∞∞ (T1,0(M)), (T1,0(M)) ⊆ (T1,0(M)).
The bundle T1,0(M) is the CR structure of M, and C∞ maps f : M → N of CR manifolds preserving the CR structures (i.e., f∗T1,0(M) ⊆ T1,0(N)) are CR maps. CR manifolds and CR maps form a category containing that of complex manifolds and holomorphic maps. The most interesting examples of CR manifolds appear, however, as real submanifolds of some complex manifold. For instance, any real hypersurface M in Cn admits a CR structure, naturally induced by the complex structure of the ambient space
T1,0(M) = T 1,0(Cn) ∩ [T (M) ⊗ C].