A generalized F-structure is a complex, isotropic subbundle E of TcM⊕Tc∗M (T_cM=TM⊗_RC and the metric is defined by pairing) such that
Nearby in the stack
E∩Eˉ⊥=0
. If
E
is also closed by the Courant bracket,
E
is a generalized CRF-structure. We show that a generalized F-structure is equivalent with a skew-symmetric endomorphism
Φ
of
TM⊕T∗M
that satisfies the condition
Φ3+Φ=0
and we express the CRF-condition by means of the Courant-Nijenhuis torsion of
Φ
. The structures that we consider are generalizations of the F-structures defined by Yano and of the CR (Cauchy-Riemann) structures. We construct generalized CRF-structures from: a classical F-structure, a pair
(V,σ)
where
V
is an integrable subbundle of
TM
and
σ
is a 2-form on
M
, a generalized, normal, almost contact structure of codimension
h
. We show that a generalized complex structure on a manifold
M~
induces generalized CRF-structures into some submanifolds
M⊆M~
. Finally, we consider compatible, generalized, Riemannian metrics and we define generalized CRFK-structures that extend the generalized Kähler structures and are equivalent with quadruples
(γ,F+,F−,ψ)
, where
(γ,F±)
are classical, metric CRF-structures,
ψ
is a 2-form and some conditions expressible in terms of the exterior differential
dψ
and the
γ
-Levi-Civita covariant derivative
∇F±
hold. If
dψ=0
, the conditions reduce to the existence of two partially Kähler reductions of the metric
γ
. The paper ends by an Appendix where we define and characterize generalized Sasakian structures.