Totally real minimal 2-spheres in quaternionic projective space

Abstract

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.