CURVATURE-ADAPTED HYPERSURFACES OF 2-TYPE IN NON-FLAT QUATERNIONIC SPACE FORMS

In a curvature-adapted hypersurface M of a quaternionic-Kähler manifold (formula presenetd) the maximal quaternionic subbundle D of TM and its orthogonal complement (formula presenetd) in TM are invariant subspaces of the shape operator at each point. We classify curvature-adapted real hypersurfaces M of non-flat quaternionic space forms (formula presenetd) and (formula presenetd) 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. The position vector of a such submanifold in the ambient (pseudo) Euclidean space is decomposable into a sum of a constant vector and two nonconstant vector eigenfunctions of the Laplace operator of the submanifold belonging to different eigenspaces. 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(formula presenetd). On the other hand, the list of 2-type curvature-adapted hypersurfaces with constant principal curvatures in (formula presenetd) is reduced to geodesic spheres and tubes of arbitrary radius about totally geodesic quaternionic hyperplane (formula presenetd). Among these hypersurfaces we determine those that are mass-symmetric or minimal. We also show that the horosphere H₃ in (formula presenetd) is not of finite type but satisfies Δ^2- _x = const.