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