Harmonic maps from spheres with lowest possible index and their properties

In this note, we prove that for n > 2 a harmonic map $\mathbb{S}^n \to \mathbb{CP}^m$, of index n+1 is pseudo horizontally weakly conformal, that $n$ must be odd. In particular, for even n every non-constant harmonic map $\mathbb{S}^n \to \mathbb{CP}^m$ has index at least n+2. We also present a harm