Rigidity of pseudo-holomorphic curves of constant curvature in Grassmann manifolds

Abstract

Rigidity of minimal immersions of constant curvature in harmonic sequences generated by holomorphic curves in Grassmann manifolds is studied in this paper by lifting them to holomorphic curves in certain projective spaces. We prove that for such curves the curvature must be positive, and that all such simply connected curves in C P n C{P^n} are generated by Veronese curves, thus generalizing Calabi’s counterpart for holomorphic curves in C P n C{P^n} . We also classify all holomorphic curves from the Riemann sphere into G ( 2 , 4 ) G(2,4) whose curvature is equal to 2 2 into two families, which illustrates pseudo-holomorphic curves of positive constant curvature in G ( m , N ) G(m,N) are in general not unitarily equivalent, constracting to the fact that generic isometric complex submanifolds in a Kaehler manifold are congruent.