We determine the Chinea-Gonzales class of almost contact metric manifolds locally realized as double-twisted product manifolds of an open interval and an almost Hermitian manifold, by means of two smooth positive functions. We also give an explicit expression for the cosymplectic defect of any manifold in the considered class and derive several consequences in dimensions 2n+1>3. Explicit formulas for two algebraic curvature tensor fields are obtained. In particular cases, this allows to state interesting curvature relations.

In the framework of Chinea-Gonzales, we study the class of almost contact metric manifolds locally realized as twisted product manifolds of an open interval and an almost Hermitian manifold, by means of a smooth positive function. Local classification theorems for the generalized Sasakian space-forms in the considered class are obtained, also.

An algebraic characterization of generalized Sasakian-space-forms is stated. Then, one studies the almost contact metric manifolds which are locally conformal to $C_6$-manifolds, simply called l.c. $C_6$-manifolds. In dimension 2n+1>=5, any of these manifolds turns out to be locally conformal cosymplectic or globally conformal to a Sasakian manifold. Curvature properties of l.c. $C_6$-manifolds are obtained, with particular attention to the k-nullity condition. This allows one to state a local classification theorem, in dimension 2n+1>=5, under the hypothesis of constant sectional curvature. Moreover, one proves that an l.c. $C_6$-manifold is a generalized Sasakian-space-form if and only if it satisfies the k-nullity condition and has pointwise constant $\varphi$-sectional curvature. Finally, local classification theorems for the generalized Sasakian-space-forms in the considered class are obtained.

Odd-dimensional non anti-invariant slant submanifolds of an α- Kenmotsu manifold are studied. We relate slant immersions into a Kähler manifold with suitable slant submanifolds of an α-Kenmotsu manifold. More generally, in the framework of Chinea-Gonzalez, we specify the type of the almost contact metric structure induced on a slant submanifold, then stating a local classification theorem. The case of austere immersions is discussed. This helps in proving a reduction theorem of the codimension. Finally, slant submanifolds which are generalized Sasakian space-forms are described.