Curvature-adapted hypersurfaces of 2-type in non-flat quaternionic space forms

We classify curvature-adapted real hypersurfaces M of non-flat quaternionic space forms HPᵐ and HHᵐ that are of Chen type 2 in an appropriately defined (pseudo) Euclidean space of quaternion-Hermitian matrices, where in the hyperbolic case we assume additionally that the hypersurace has constant principal curvatures. In the quaternionic projective space they include geodesic hyperspheres of arbitrary radius r ∈ (0, π/2) except one, two series of tubes about canonically embedded quaternionic projective spaces of lower dimensions and two particular tubes about a canonically embedded CPᵐ ⊂ HPᵐ. On the other hand, the list of 2-type curvature-adapted hypersurfaces with constant principal curvatures in HHᵐ is reduced to geodesic spheres and tubes of arbitrary radius about totally geodesic quaternionic hyperplane HHᵐ⁻¹. Among these hypersurfaces we determine those that are mass-symmetric or minimal. We also show that the horosphere H₃ in HHᵐ is not of finite type but satisfies Δ² x = const.