CR-submanifolds of \(\mathbb{H} P^m\) and hypersurfaces of the Cayley plane whose Chen-type is 1 (Q2718055)

A submanifold \(M\) of Euclidean space \(\mathbb R^N\) is of Chen type \(1\) iff up to constants, all its coordinate functions are eigenfunctions of the Laplacian on \(M\) with the same eigenvalue. Submanifolds of \(\mathbb HP^m\) are identified with submanifolds of the Euclidean space of Hermitian quaternionic \((m+1)\)-matrices by a type \(1\) embedding \(\mathbb HP^m\subset\mathbb R^N\); for the Cayley plane, the Freudenthal embedding \(\mathbb OP^2\subset\mathbb R^N\) is used instead. A submanifold of \(\mathbb HP^m\) or \(\mathbb OP^2\) is defined to be of Chen type \(1\) iff its image in \(\mathbb R^N\) is of Chen type \(1\). NEWLINENEWLINENEWLINEA submanifold \(M\subset\mathbb HP^m\) is called quaternion CR if \(TM=\mathcal D\oplus\mathcal D^\perp\) such that \(J\mathcal D\subset TM\) and \(J\mathcal D^\perp\subset T^\perp M\) for all local almost complex structures \(J\) given by the quaternionic Kähler structure \(\mathcal J\) on \(\mathbb HP^m\). \(M\) is called anti-CR if \(T^\perp M\) admits a similar decomposition. \(M\subset\mathbb HP^m\) is called totally complex iff \(TM\) is preserved by one parallel almost complex structure \(J\in\mathcal J|_M\) whereas the local almost complex structures in \(J^\perp\subset\mathcal J|_M\) map \(TM\) into \(T^\perp M\). NEWLINENEWLINENEWLINEThe author classifies all Chen type \(1\) submanifolds of \(\mathbb HP^m\) that are either quaternion CR or anti-CR or totally complex. Among the possibilities are the totally geodesic submanifolds \(\mathbb HP^n\), \(\mathbb CP^n\) for \(n\leq m\), minimal totally real or anti-Lagrangian submanifolds, and geodesic spheres of radius \(\cot^{-1}\sqrt{3/(4n+1)}\) in a totally geodesic \(\mathbb HP^n\). The author also proves that the only Chen type \(1\) hypersurfaces of \(\mathbb OP^2\) are totally geodesic spheres of radius \(\cot^{-1}\sqrt{7/17}\).