We study isometric immersions into an almost contact metric manifold which falls in the Chinea-Gonzalez class C5 ⊕ C12, under the hypothesis that the Reeb vector eld of the ambient space is normal to the considered submanifolds. Particular attention to the case of a slant immersion is paid. We relate immersions into a Kähler manifold to suitable submanifolds of a C5 ⊕ C12-manifold. More generally, in the framework of Gray-Hervella, we specify the type of the almost Hermitian structure induced on a non anti-invariant slant submanifold. The cases of totally umbilical or austere submanifolds are discussed.
We investigate slant surfaces in the almost Hermitian manifold 𝕊3 × ℝ, considering the position of the Reeb vector field ξ of the Sasakian structure on 𝕊3 with respect to the surfaces. We examine two cases: ξ normal or tangent to the surfaces. In the first case, we prove that every surface is totally real. In the second case, we characterize and locally describe complex surfaces. Finally, we completely classify non-complex slant surfaces, giving explicit examples.
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