Suppose that $$S^n$$ S n is given a generic Riemannian metric with sectional curvatures which satisfy a suitable pinching condition formulated in terms of complex sectional curvatures. This pinching condition is satisfied by manifolds whose real sectional curvatures $$K_r(\sigma )$$ K r ( σ ) satisfy $$\begin{aligned} 1/2 < K_r(\sigma ) \le 1. \end{aligned}$$ 1 / 2 < K r ( σ ) ≤ 1 . Then the number of minimal two spheres of Morse index $$\lambda $$ λ , for $$n-2 \le \lambda \le 2n-5$$ n - 2 ≤ λ ≤ 2 n - 5 , is at least $$p_{3}(\lambda -n+2)$$ p 3 ( λ - n + 2 ) , where $$p_{3}(k)$$ p 3 ( k ) is the number of k-cells in the Schubert cell decomposition for $$G_3({\mathbb {R}}^{n+1})$$ G 3 ( R n + 1 ) .