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Suppose that (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 )) satisfy $$begin{aligned} 1/2 u003c K_r(sigma ) le 1. end{aligned}$$Then the number of minimal two spheres of Morse index (lambda ), for (n-2 le lambda le 2n-5), is at least (p_{3}(lambda -n+2)), where (p_{3}(k)) is the number of k-cells in the Schubert cell decomposition for (G_3({mathbb {R}}^{n+1})).

Minimal two-spheres of low index in manifolds with positive complex sectional curvature