Quaternionic distribution of curvature-adapted real hypersurfaces in a quaternionic hyperbolic space (Q1396026)

A hypersurface \(M\) of a Riemannian manifold \(\widetilde M\) is called curvature-adapted if the normal Jacobi operator \(K\) and the shape operator \(A\) of \(M\) with respect to a unit normal vector field \(\mathcal N\) satisfy \(K\circ A=A\circ K\). In the reviewed paper the authors consider curvature-adapted real hypersurfaces \(M\) in a quaternionic \(n\)-dimensional hyperbolic space \(\mathbb H^n\). The quaternionic distribution \(\mathcal D\) of \(M\) is the maximal subbundle of \(TM\) which is invariant by the quaternionic structure of \(\mathbb H^n\). It is proved that \(\mathcal D\) is not integrable and therefore the authors study some integrability conditions related to some subdistributions of \(\mathcal D\). They give a characterization of all curvature-adapted real hypersurfaces \(M\) with constant principal curvatures in \(\mathbb H^n\) by investigating integrability conditions for some distributions by means of principal distributions contained in \(\mathcal D\).