Totally real minimal 2-spheres in quaternionic projective space

Let HPn be the quaternionic projective space with constant quaternionic sectional curvature 4. Then locally there exists a tripe {I, J, K} of complex structures on ℍPn satisfying IJ = -JI = K,JK = -KJ = I,KI = -IK = J. A surface M ⊂ ℍPn is called totally real, if at each point p ∈ M the tangent plane TpM is perpendicular to I(TpM), J(TpM) and K(TpM). It is known that any surface M ⊂ ℝPn ⊂ ℍPn is totally real, where ℝPn ⊂ ℍPn is the standard embedding of real projective space in ℍPn induced by the inclusion ℝ in ℍ, and that there are totally real surfaces in ℍPn which don’t come from this way. In this paper we show that any totally real minimal 2-sphere in ℍPn is isometric to a full minimal 2-sphere in ℝP2m ⊂ ℝPn ⊂ ℍPn with 2m ≤ n. As a consequence we show that the Veronese sequences in ℝP2m (m ≥ 1) are the only totally real minimal 2-spheres with constant curvature in the quaternionic projective space.